In Algebra Vol. 2 the author, P. M. Cohn writes the following in chapter 7.7
Let $H$ be a subgroup of finite index $r$ in $G$ and consider a right $H$-module $U$. From it we can form a right $G$-module $$\tag{1} \text{ind}^G_H=U^G=U\bigotimes_HkG, $$ called the induced $G$-module.
My question now is, what exactly does the $H$ in $\bigotimes_H$ mean? In chapter 4.7 he defines $U\otimes_K V$ as the tensor product of two $K$-modules $U$ and $V$, where $K$ is a commutative ring, but I do not see how I 'translate' this notation to that in equation $(1)$ where $U$ is an $H$-module and $kG$ a $G$-module with $H$ and $G$ being groups.
Edit: I have an additional doubt. He goes on to write
To find the representation afforded by $U^G$ we note that the subspace $U\otimes H=\{u\otimes 1|u\in U\}$ of $U^G$ is an $H$-module in a natural way: for any $h\in H$ we have $(u\otimes 1)h=uh\otimes 1$. More generally we can define the subspace $U\otimes Ha=\{u\otimes a|u\in U\}$ of $(1)$ as an $H^a$, where $H^a=a^{-1}Ha$, by the rule $$ (u\otimes a)x=uaxa^{-1}\otimes a,\quad \text{for } x\in H^a. $$
Here I do not understand why he writes $U\otimes H$, which appears to denote the tensor product between an $H$-module, $U$, and a group $H$?