Suppose that I have a morphism of real algebraic groups $H({\mathbb R}) \to G({\mathbb R})$ with finite, abelian cokernel, and suppose that both groups admit discrete series representations. Let $\pi$ be a discrete series representation of $G({\mathbb R})$ and consider the representation of $H({\mathbb R})$ on $\pi$ by simply restricting it to $H({\mathbb R})$. My question is, which should be simple, can one explicitly describe the representations of $H({\mathbb R})$ which occur in this restriction?

I do not mind to put additional conditions, such as perhaps a regularity condition on the representation $\pi$. (But preferably no strong conditions on the groups, edit: other than ``reductive'').

EDIT: More specifically I need to know if, $\pi$ is a discrete series representation of $G({\mathbb R})$ which is sufficiently regular in some sense, does then the restriction of $\pi$ to $H(\mathbb R)$ consist of only sufficiently regular representations?