Let $G$ be a group $H\leq G$ a subgroup of finite index. Further, let ${\mathcal E}^G_H$ denote the class of those short exact sequences of $G$-modules (over some fixed base ring) which split when regarded as sequences of $H$-modules. Then $(G\text{-mod},{\mathcal E}^G_H)$ is a Frobenius category.

Does anybody know if it is true that $(G\text{-mod},{\mathcal E}^G_H)$ is *not* Frobenius if $(G:H)=\infty$?

To show this, it would be enough to construct for and pair $(G,H)$ with $(G:H)=\infty$ an $H$-module $M$ such that the canonical $H$-split map

$\text{Ind}^G_H M\hookrightarrow\text{Coind}^G_H\text{Res}^G_H\text{Ind}^G_H M$

is not $G$-split. For example, if $M$ is the trivial $H$-module, this map becomes

${\mathbb Z}[G/H]\hookrightarrow\text{Hom}_{{\mathbb Z}H}({\mathbb Z}G,{\mathbb Z}[G/H])$

$g_0 H\longmapsto (g\mapsto g g_0 H)$.

Unfortunately, I'm not able to proof that this map is not $G$-split.