Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, for example in Marko Tadić's paper Notes on representations of non-Archimedean $\operatorname{SL}(n)$. Consider the following theorem:

**Theorem**: Let $\pi_1$ be an irreducible admissible representation of $G_1$. There exists an irreducible admissible representation $\pi$ of $G$ such that $\pi_1$ is isomorphic to a subrepresentation of $\pi\rvert_{G_1}$. If $\pi_1$ is generic and supercuspidal, then $\pi$ can be chosen to be generic and supercuspidal.

The proof involves finding a finite rank free group $S_1$ inside $Z(G)$ such that $S_1 \cap G_1 = 1$, and $G/S_1G_1$ is compact. One extends $\pi$ to $S_1G_1$ by making it trivial on $S_1$, and then induces to $G$.

I was wondering how much control we may have for the central character. If $\pi_1$, $\pi_1'$ are irreducible admissible representations of $G_1$ having the same central character, is it always possible to choose the $\pi$, $\pi'$ in the theorem to also have the same central character? If anyone knows a reference that deals with this sort of question, it would be greatly appreciated.