How to distinguish two "almost isomorphic" groups I'd like to ask the following:
Let $G$ and $H$ be the two finite groups, which are not isomorphic, but "almost", i.e. they share a lot of common properties.
Let $G:=$SmallGroup(605,5) and $H:=$SmallGroup(605,6).

How can you distinguish $G$ and $H$ in this case without just saying that IdSmallGroup($G$)$\neq$IdSmallGroup($H$)?

Of course, you could say that $G\cong H$, iff their group algebras are isomorphic as Hopf algebras.
But, concretely, I am searching for an invariant under isomorphism to distinguish them, such like $G'\not\cong H'$ or $G$ is abelian and $H$ is not or the number of conjugacy classes is different or the Frattini subgroups are not isomorphic, etc.
Unfortunately, I couldn't find any invariant which distinguishes them by now.
I would be thankful for any help.
EDIT(10th March 2020): I changed the question, because it was closed for the following reason: This question needs to be more focused.
 A: It is a well-known fact that solvable groups are difficult to classify because of the abundance of normal subgroups. In a certain sense the class of groups of primepower order is the simplest case to handle. But already here the classification is far from being complete. Up to now there are only a few classes of such primepower order groups which have been adequately analysed. 
The first to create
some order in the plethora of groups of prime-power order was Philip Hall. He
observed that the notion of isomorphism of groups is really too strong to give
rise to a satisfactory classification and that it had to be replaced by a weaker
equivalence relation. Subsequently he discovered a suitable equivalence relation
and called it isoclinism of groups (see Ph. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130-141.). It is this classification principle that underlies the famous monograph of M. Hall and J. K. Senior on the classification of 2-groups of order at most 64.The isoclinism class of a group G is determined by the groups $G/Z(G)$ (the inner automorphism group) and $G′$ (the commutator subgroup) and the commutator map from $G/Z(G) \times G/Z(G)$ to $G’$. In other words, two groups $G_1$ and $G_2$ are isoclinic if there are isomorphisms from $G_1/Z(G_1)$ to $G_2/Z(G_2)$ and from $G_1′$ to $G_2′$ commuting with the commutator map. See also here.
