A note on semidirect products and nonabelian tensor products of groups

Let $G$ and $H$ be groups which act compatibly on one another. In [2] and [8] it is considered a group construction $\eta(G,H)$ which is related to the nonabelian tensor product $G\otimes H$. In this note we study embedding questions of certain semidirect products $A\rtimes H$ into $\eta(A, H)$, for finite abelian $H$-groups $A$. As a consequence of our results we obtain that complete Frobenius groups and affine groups over finite fields are embedded into $\eta(A, H)$ for convenient groups $A$ and $H$. Further, on considering finite metabelian groups $G$ in which the derived subgroup has order coprime with its index we establish the order of the nonabelian tensor square of $G$.