Question: For each of the groups $P$ of order 16 determine the classes of subgroups of index $2$ under the action of $\operatorname{Aut}(P)$.
Context: For almost $60$ years it has seemed to me that on their $N$-th birthday group theorists ought to classify (or at least know something about) the groups of order $N$. After a long run of moderately easy years I will soon be faced with the case $N=80$. I know that there are $52$ such groups but want to "understand" them.
If $|G|=60$, $P\in\operatorname{Syl}_2(G)$, $Q\in\operatorname{Syl}_5(G)$ then we have to deal with the following cases.
(A) $Q\not\lhd G$. In this case $G$ must be the semidirect product of $\mathbb{Z}_2^4$ by a $5$-cycle acting as the companion matrix of $X^4+X^3+X^2+X+1$.
(B) $Q\lhd G$ and $N(Q)/C(Q)=1$. Here we get the direct products of each of the $14$ groups of order $16$ with $C_5$.
(C) $Q\lhd G$ and $N(Q)/C(Q)\simeq C_2$. Answers to the question posed will let us describe which semidirect products occur.
(D) $Q\lhd G$ and $N(Q)/C(Q)\simeq C_4$. The answer to Groups of order $16$ with a cyclic quotient of order $4$ by @DerekHolt lets us describe which semidirect products occur.
My Working: Using this list https://groupprops.subwiki.org/wiki/Groups_of_order_16 we can see (by considering the Frattini quotient) that we have the following number of subgroups of index 2 for each group.
group 1 : One (and it is unique)
groups 2-9: Three each (and by looking at isomorphism types there must be at least two classes under the automorphism group)
groups 10-13: Seven each
group 14: Fifteen (but all equivalent under the automorphism group)
(I note that subgroups of index $2$ are normal, so that one only needs to know the action of the outer automorphism group of $P$.)