About using conjugation in $S_4$ to help in finding the outer automorphisms of $D_4$ I'm studying the ${\rm Aut}(D_n)$ from other posts here, even if not all of them are so easy to me but I'm trying to collect them all and get a good view about them.
It's clear ${\rm Inn}(D_4)\cong D_4/Z(D_4)$, which leads to to a group isomorphic to $V_4$.
So there will be then remaining four outer automorphisms leading to $Aut(D_4)\cong D_4$.
Considering $D_4$ is isomorphic to a subgroup of $S_4$, I was wondering if using the conjugation of $D_4$ in $S_4$ could give any hints to estimate the other four outer automorphisms.
It's just an intution, but I may be wrong of course.
Thanks in advance.
 A: I think conjugations in $S_4$ will not help to find all automorphisms of the group $D_4$. But to calculate all automorphisms it is indeed useful to know that the subgroup
$$
H=\operatorname{gr}((1324),(12)(34))\cong D_4.
$$
For the group $H$ there are the following $4$ pairs of generating involutions:
\begin{eqnarray*}
 (12),(13)(24);\\
(12),(14)(23);\\
(34),(13)(24);\\
(34),(14)(23).
 \end{eqnarray*}
Since every automorphism of the group $H$ is obviously uniquely defined if we know the image of any of these pairs.
Here are 8 automorphisms:
\begin{eqnarray*}
\phi_1(12)=(12),\phi_1((13)(24))=(13)(24);\\
\phi_2(12)=(12),\phi_2((13)(24))=(14)(23);\\
\phi_3(12)=(34),\phi_3((13)(24))=(13)(24);\\
\phi_4(12)=(34),\phi_4((13)(24))=(14)(23);\\
\phi_5(12)=(13)(24),\phi_5((13)(24))=(12);\\
\phi_6(12)=(14)(23),\phi_6((13)(24))=(12);\\
\phi_7(12)=(13)(24),\phi_7((13)(24))=(34);\\
\phi_8(12)=(14)(23),\phi_8((13)(24))=(34);\\
\end{eqnarray*}
The first four automorphisms are inner, the last four are outer.
