The number of permutational products of two finite groups

Let $H$ be a subgroup of the two finite groups $A$ and $B$. Let $\mathcal {S}\;(\mathcal {J})$ be the set of transversals of the left cosets of $H$ in $A\;(B)$, and consider $A\;(B)$ as a permutation group on $\mathcal {S}\;(\mathcal {J})$ where the action is by left multiplication. If ${n_A}\;({n_B})$ is the number of orbits under the action of $A\;(B)$ then the number of nonisomorphic permutational products of $A$ and $B$ amalgamating $H$ is bounded by ${n_A}{n_B}$.