  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.
