Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup

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 $\mathrm{\mid }P\mathrm{\mid }\ge 8$, $TT(G)\cong X(G)$ the group of one-dimensional $kG$-modules, except possibly when $G\mathrm{/}{O}_{2^{\mathrm{\prime }}}(G)\cong {\mathfrak{A}}_6$, 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 $\mathrm{\mid }P\mathrm{\mid }=4$, 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 $TT(G\mathrm{/}H)$, where $H$ is a normal $p^{\mathrm{\prime }}$-subgroup of $G$.