**Context:**
Let $\pi: \widehat{G} \rightarrow G$ be a surjective morphism between connected reductive groups defined over $\mathbb{F}_q$ whose kernel is a central torus. Then $\pi : \widehat{G}^F \rightarrow G^F$ is surjective, where $F$ denotes the Frobenius morphisms inducing the $\mathbb{F}_q$-rational structures.
We thus have an inflation map $- \circ \pi: \text{Irr}(G^F) \rightarrow \text{Irr}(\widehat{G}^F)$. We also have a dual morphism $\pi^* : G^* \rightarrow \widehat{G}^*$ which is injective under the above assumptions.

**Question:** Is the inflation map compatible with Lusztig series in the sense that for a semisimple $s \in (G^*)^F$ and any $\chi$ in the Lusztig series $\mathcal{E}(G^F,[s])$ associated to $s$ we have $\chi \circ \pi \in \mathcal{E}(\widehat{G}^F,[\pi^*(s)])$?

**Motivation:**
It is easy to see that every element of $\mathcal{E}(\widehat{G}^F,[\pi^*(s)])$ is inflated from $G^F$, so if the answer to my question is positve, this would result in a bijection between $\mathcal{E}(G^F,[s])$ and $\mathcal{E}(\widehat{G}^F,[\pi^*(s)])$.

**My Approach:**
It seems natural to me to try to show that inflation is compatible with Lusztig induction, i.e. $(R^G_T \theta) \circ \pi = R^{\widehat{G}}_{\pi^{-1}(T)} (\theta \circ \pi)$ for any rational maximal torus $T$ and any $\theta \in \text{Irr}(T^F)$. To show this one could try to compare the Deligne-Lusztig varieties $Y_U = \{g \in G \:|\: g^{-1}F(g) \in U\}$ and $Y_{\pi^{-1}(U)}$ for $U$ the unipotent radical of a Borel subgroup of $G$ containing $T$.
This looks promising to me as $\pi$ induces a surjective morphism $Y_{\pi^{-1}(U)} \rightarrow Y_{U}$ whose fibres are cosets of $\ker(\pi)^F$. As I see it it would suffice to show that this morphism gives an isomorphism between the $\ker(\pi)^F$-quotient of $Y_{\pi^{-1}(U)}$ and $Y_{U}$ but I am unable to prove it.

I tried to mimic the proof of Proposition 13.20 in Representations of Finite Groups of Lie Type by Digne and Michel where the case $s = 1$ is studied for a more general $\pi$. But unfortunately and contrary to what is claimed there the proof given there does not establish an isomorphism like the above as far as I can tell. Any help is appreciated.