Continuing from Martin's answer, here is a longer list of properties that are preserved by any auto-equivalence of FinGrp:
- The subgroup lattice is preserved, in particular infima (=intersections) and suprema (=subgroup generated by smaller subgroups) are preserved.
- As observed the normal structure is preserved: $H\hookrightarrow G$ is normal iff it is the difference kernel of some arrow $G\to X$ with the trivial arrow $G\to X$. Quotients are preserved by the universal property of quotients.
- In particular nilpotent (sylowgroups are normal), supersolvable ($\exists$ normal series with cyclic quotients), solvable ($\exists$ subnormal series with abelian quotients) groups are preserved.
- For solvable groups we can say more: The commutator group is preserved (minimal normal subgroup with abelian quotient), hence the derived series is preserved.
- The center is the maximal normal subgroup $Z\leq G$ such that $ZH$ is abelian for all cyclic subgroups $H\leq G$. Therefore upper and lower central series are preserved.
- Normal subgroups defined in terms of these notions like $O_p$, $O_\pi$, $O^p$, $O^\pi$, the Fitting subgroup $F(G)$ etc. are preserved.
- Since centers are preserved, central extensions and Schur multipliers are preserved. In particular quasisimple groups (=perfect, central extension of simple) are preserved.
- Therefore components (= subnormal and quasisimple subgroups) are preserved and therefore the generalized Fitting subgroup $F^\ast(G)$.
The Frattini subgroup (=intersection of all maximal subgroups) is preserved.
The centralizer $C_G(H)$ is the subgroup generated by all cylic subgroups $\langle z\rangle\leq G$ such that $\langle z\rangle$ and $\langle h\rangle$ generate an abelian subgroup for all cyclic subgroups $\langle h\rangle \leq H$.
- This should imply that the isomorphism class of each simple group is fixed. I haven't checked all details though.
This suggests that the "non-solvable part" of group theory is completely preserved. Things may get way a lot more complicated for $p$-groups since there are too many groups and too few invariants to really distinguish them. Well let's start anyway:
- Order, class, coclass are preserved
- For groups of small coclass there is a classification project by Bettina Eick, Max Horn et.al. It should imply that isomorphism classes of 2-groups of coclass $\leq 3$ are fixed etc.
EDIT 1:
Lemma: The isomorphism class of every finite Coxeter group is fixed.
Proof: Remember a Coxeter group is a group $W$ that is generated by a set of involutions $S\subseteq W$ (=a set of subgroups of size 2) modulo the relations $(st)^{m_{st}}=1$ for some $m_{st}=m_{ts}\in\mathbb{N}$ (with $m_{ss}=1$ of course, otherwise s wouldn't be an involutions). These relations can be equivalently rephrased as "$W$ is generated by cyclic subgroups $\langle s\rangle$ of order 2, $\langle s\rangle$ and $\langle t\rangle$ generate a group of size $2m_{st}$ for all $s,t$ and $W$ is maximal with this property". With this description it is clear that any Coxeter group is mapped to a Coxeter group.
In fact: If $\{\langle s\rangle\to W | s \in S\}$ is a Coxeter generating set, then the unique (!) isomorphism $\langle s\rangle \to \mathcal{F}(\langle s\rangle)$ extends to an isomorphism $W\to\mathcal{F}(W)$ for every auto-equivalence $\mathcal{F}:FinGrp\to FinGrp$ and these isomorphisms are compatible with parabolic subgroups.
In particular we get a sequence of isomorphisms $Sym(n)\to\mathcal{F}(Sym(n))$ that commute with the standard embeddings $Sym(n)\to Sym(n+1)$.
EDIT 2:
Lemma 2: (Inner) Automorphism groups (as object within FinGrp, not as morphism sets!) are preserved.
Proof: Fix $N$ and consider splitting, exact sequences $1\to N\to G \leftrightarrows A\to 1$. Every such sequence $S$ induces a morphism $\kappa_S : A\to Aut(N)$. Now consider all morphisms (=arrows $N\to N$, $\alpha:G\to G'$, $\beta:A\to A'$ such that everything commutes) between such sequences $S,S'$ where $N\to N$ is the identity. One verifies that $\kappa_{S'}(\beta(a)) = \kappa_S(a)$ for all $a\in A$.
The sequence $1\to N\to N\rtimes Aut(N)\to Aut(N)\to 1$ is a terminal object in the category of these sequences. The unique morphism from every seqence to it is induced by the $\kappa_S$ as one easily verifies.
A sequence $1\to N\to G\to A\to 1$ is a sequence with $A\leq Inn(N)$ iff $G=N\cdot C_G(N)$.
QED.
EDIT 3:
I think we're almost there. For every prime $\ell$ the category of finite dimensional $\mathbb{F}_\ell[G]$-modules is preserved by autoequivalences:
A $\mathbb{F}_\ell[G]$-module is the same thing as a split exact sequence $1\to V\to X\leftrightarrows G\to 1$ with $V$ abelian and $\ell$-torsion. A $G$-linear map $V\to W$ is the same thing as a morphism between the two sequences which is the identity on $G$. If one drops the condition of $\ell$-torsion and takes projective limits one similarly describes the categories of finitely generated $\mathbb{Z}_\ell[G]$-modules.
We therefore know that all these group rings for $G$ and $\mathcal{F}G$ are morita equivalent.
In fact this shows something stronger: Every autoequivalence $\mathcal{F}:FinGrp\to FinGrp$ induces an equivalence $\mathbb{F}_\ell[G]-mod\to\mathbb{F}_\ell[\mathcal{F}G]-mod$ which preserves the underlying vector spaces (since we already know that $\mathcal{F}$ is equivalent to the identity on abelian groups). The nLab entry on Tannaka duality tells us that this is enough to obtain an isomorphism of the group rings which should be (I haven't checked all the details) natural.
Additionally the trivial representation in each of these categories is preserved: it is the sequence $1\to V\to X\leftrightarrows G\to 1$ with $V\cong C_p$ that has a retract $X\to V$ such that $1\leftarrow V\leftarrow X\leftarrow G\leftarrow 1$ is also exact. Does anyone see how to characterize dual representations and tensor products of representations? In that case we could get the isomorphism for the groups an not just the group rings.