Let $G$ be a reductive group over the finite field $\mathbb{F}_q$. Then a *regular* embedding of $G$ is an $\mathbb{F}_q$-rational embedding $\iota \colon G \rightarrow G'$ into a second reductive group over $\mathbb{F}_q$, such that $\iota$ maps the derived group of $G$ isomorphically onto that of $G'$ and the center of $G'$ is connected. Moreover, a regular embedding is called *smooth regular*, if additionally the center of $G'$ is smooth. It is well-known that smooth regular embeddings always exist (see for example [1,Theorem 4.5],[2, Lemma 6.5]).

**Question.** Is it true that for any $G$ a smooth regular embedding $\iota \colon G \rightarrow G'$ exists, with the additional property that the center of $G'$ is an induced torus?

It seems quite plausible, but I could not find any reference (which is a bit annoying).

Note that for the somewhat dual notion of a *z-extension* $\widetilde G \rightarrow G$ it can be arranged -- and is even part of the definition of a z-extension (cf. [3,§1]) -- that ${\rm ker}(\widetilde{G} \rightarrow G)$ is induced.

References:

[1] Martin, B.: Étale slices for representation varieties in characteristic p.

[2] Taylor, J.: The Structure of Root Data and Smooth Regular Embeddings of Reductive Groups.

[3] Kottwitz, R.: Rational conjugacy classes in reductive groups.