Let $V_n\simeq V_m^{1} \otimes V_m^{2} \cdots \otimes V_m^{k}$ be an $n=m^k$-dimensional vector space over some field $F$, and $\textrm{GL}_n(F)$ the general linear group over that vector space. The superscripts are here only for bookkeeping. Let $H$ be the subgroup generated by permutations of the $V_m^i$ factors and elements $g$ in $\textrm{GL}_n(F)$ that act only on the factors. That is, $g = g_1 \otimes g_2 \otimes \cdots g_k$, where $g_i \in \textrm{GL}_n^i$. Such subgroups naturally arise in cases where the large group acts on a set that is invariant under `local operations' and permutations.
Is there a name for such a subgroup, or any references for this? Is $H$ an outer semidirect product between $\textrm{GL}_m^{1}\times \textrm{GL}_m^2 \times\cdots \textrm{GL}_m^{k}$ and a group isomorphic to $S_n$? Are such subgroups maximal?
One can also make a similar construction, but with $\textrm{GL}_n(F)$ replaced by $\textrm{SL}_n(F)$, $\textrm{SP}_{2n}(F)$ and similiar groups (think $S_n$), and/or replacing the tensor product with the direct product.