Given a finite permutation group $G$ and an element $a\in G$ with conjugacy class $X$, I am interested in determining when for a given element $x\in X$ the subgroup $<a,x>$ generated by $a$ and $x$ is isomorphic to a fixed subgroup $H\leq G$.
To identify which elements of $X$ satisfy this property, I am using the computer algebra software MAGMA, with the command IsIsomorphic(sub$<G|a,x>$,$H$). Naturally I can improve the efficiency of the algorithm by not naively checking this isomorphism for all pairs $(a,x)\in X\times X$, but rather removing an element from $X$ once I have checked whether it generates $H$ with any other element that has not already been checked (I hope that makes sense). However, this still seems to be an inefficient way to proceed.
Given that I know $\vert H\vert$ and I also know $\vert ax\vert$ in the case that $<a,x>\cong H$, then does anyone know whether it will be more efficient to check these properties first before using the command IsIsomorphic or will it be less efficient (since I am assuming that these checks will be built into the IsIsomorphic command).