Let $G$ be a finite group of length $\ell(G)=n$, if $H$ is a decomposition series for $G$, we denote by $L(H)$ the ordered set of its factors.
I'm trying to figure out if the set $X_G:=\{\sigma\in\mathfrak{S}_n$| there exist $H,F$ two composition series for $G$ such that $L(F)=\sigma(L(H))\}$ is a group under some (resonable) additional conditions.
It's clear that identity is an element of $X_G$ and that $\forall\sigma\in X_G$, $\sigma^{-1}\in X_G$; the real problem is to establish when $X_G$ is closed under permutation product.
I haven't found any counter example yet.
Edit: @DerekHolt pointed out that, actually, the group $C_6\times\mathfrak{S}_3$ is a counter example. I also noticed that for any finite abelian group $G$ of length $\ell(G)=n$, $X_G=\mathfrak{S}_n$. Moreover if $m$ is odd, then $X_{D_m}=\mathfrak{S}_{\ell(D_m)-1}$, where $D_m$ is the $m$-th dihedral group.