While working on a problem, I constructed something which looked like an induced representation, but with a tensor product instead of a direct sum.

Here is a special case. Let $G$ be a group, with $H$ a subgroup of index $2$. Choose $s \in G$ which is not in $H$. For $(\pi,V)$ a representation of $H$, define a representation $\sigma$ of $G$ with underlying space $V \otimes V$ as follows: if $h \in H$, define $\sigma(h) = \pi(h) \otimes \pi(shs^{-1})$. If $g \in G, \not\in H$, define $\sigma(g)$ on generators by

$$\sigma(g) v \otimes w = \pi(gs^{-1})w \otimes \pi(sg)v$$

If we had used $V \oplus V$ instead of $V \otimes V$, then following the above construction would have produced the representation $\operatorname{Ind}_H^G(\pi)$. So this is like a tensor product version of induced representation.

I found some papers which talk about "tensor induction," and I believe the above construction is a very simple case of that. What is tensor induction, generally speaking? Is it adjoint to some functor, or satisfy a universal property?