  Let $k$ be an algebraically closed field of characteristic $p>0$ and $G$ a
finite group. We provide a description of the torsion subgroup $TT(G)$ of the
finitely generated abelian group $T(G)$ of endo-trivial $kG$-modules when $p=2$
and $G$ has a dihedral Sylow $2$-subgroup $P$. We prove that, in the case
$|P|\geq 8$, $TT(G)\cong X(G)$ the group of one-dimensional $kG$-modules,
except possibly when $G/O_{2'}(G)\cong \mathfrak{A}_6$, the alternating group
of degree $6$; in which case $G$ may have $9$-dimensional simple torsion
endo-trivial modules. We also prove a similar result in the case $|P|=4$,
although the situation is more involved. Our results complement the
tame-representation type investigation of endo-trivial modules started by
Carlson-Mazza-Th\'evenaz in the cases of semi-dihedral and generalized
quaternion Sylow 2-subgroups. Furthermore we provide a general reduction
result, valid at any prime $p$, to recover the structure of $TT(G)$ from the
structure of $TT(G/H)$, where $H$ is a normal $p'$-subgroup of $G$.
