Let H, N be two groups with H cyclic. Let $f,g:H \rightarrow Aut(N)$ be homomorphisms such that $N\rtimes _f H \cong N\rtimes _g H$. Then does $f(H)$ and $g(H)$ are conjugate in $Aut(N)$?
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

The answer is no. You can easily have a situation where $f$ is the trivial map, while $g$ makes $H$ act through an inner automorphism of $N$, so that in both cases $N\rtimes H\cong N\times H$. For concreteness, let $N=D_8$ be the dihedral group of order 8, let $\sigma$ be a noncentral involution in $N$, let $H=\langle h\rangle\cong C_2$, and define $g(h)(n)=\sigma^{1} n\sigma\;\forall n\in N$. Then $G=N\rtimes_g H\cong N\times H$, since the subgroup $\langle h\sigma\rangle$ of $G$ is of order 2, intersects $N$ trivially, and commutes with it. 

