Let Q ≤ P be finite p-groups, H ≤ Aut(Q).
Is it really true that there is at most one p-solvable group G such that $Q \unlhd G$, $C_G(Q) \leq Q$, P is a Sylow p-subgroup of G, and the map from G to Aut(Q) surjects onto H?
I think this is true (up to an isomorphism of G restricting to the identity on Q), but I am worried about a consequence:$\newcommand{\Aut}{\operatorname{Aut}}\newcommand{\Fit}{\operatorname{Fit}}$
Let F be a finite nilpotent group, let H be a subgroup of Aut(F), and for each prime p dividing the order of F, let Fp ≤ Ep be a Sylow p-subgroup of F contained in some finite p-group.
Say that G is a model of $(H,E)$ if G is solvable, $F \unlhd G$, $C_G(F) \leq F$, the homomorphism from G to $\Aut(F)$ is surjective onto H, and each Ep is a Sylow p-subgroup of G.
Is it really true that given F, H, and E there is at most one (up to an isomorphism restricting to the identity on F) model of $(H,E)$?
In such a model, F is the largest nilpotent normal subgroup of G, and a theorem of Fitting guarantees that $G/Z(F) \cong H$, so of course $G/Z(F)$ is uniquely determined by H. I had no idea G itself could be uniquely recovered if only ones knows the Sylow subgroups (assuming I am not wrong).
If it is false, I would appreciate an example where F = Q is a p-group.
If it is true, I would appreciate an older reference than 21st century topology, as surely this is “Fitting's other theorem.”
A special case is clear: if F has vanishing second cohomology as an H-module, then G must be the semi-direct product. In particular, if the orders of F and H are coprime, then of course G is uniquely determined.
On the other hand, if the second cohomology does not vanish, then while multiple extensions can arise, in the examples I've seen, each extension is uniquely identified by its Sylow subgroups that intersect the Fitting subgroup non-trivially.
I am not sure I understand how knowing the Sylows, but not knowing how they interact is sufficient to know the group. A good answer (if it is true) might begin “but Jack we do know how they interact, because…”.