Endo-trivial modules for finite groups with Klein-four Sylow 2-subgroups

We study the finitely generated abelian group $T(G)$ of endo-trivial $kG$-modules where $kG$ is the group algebra of a finite group $G$ over a field of characteristic $p>0$. When the representation type of the group algebra is not wild, the group structure of $T(G)$ is known for the cases where a Sylow $p$-subgroup $P$ of $G$ is cyclic, semi-dihedral and generalized quaternion. We investigate $T(G)$, and more accurately, its torsion subgroup $TT(G)$ for the case where $P$ is a Klein-four group. More precisely, we give a necessary and sufficient condition in terms of the centralizers of involutions under which $TT(G) = f^{-1}(X(N_{G}(P)))$ holds, where $f^{-1}(X(N_{G}(P)))$ denotes the abelian group consisting of the $kG$-Green correspondents of the one-dimensional $kN_{G}(P)$-modules. We show that the lift to characteristic zero of any indecomposable module in $TT(G)$ affords an irreducible ordinary character. Furthermore, we show that the property of a module in $f^{-1}(X(N_{G}(P)))$ of being endo-trivial is not intrinsic to the module itself but is decided at the level of the block to which it belongs.