The Alperin theorem for periodic groups with a finite Sylow $2$-subgroup

We transfer the well-known Alperin theorem on fusion of the $p$-elements of Sylow $p$-subgroups of finite groups onto periodic groups with finite Sylow $2$-subgroups for the case $p=2$. The basis for this transfer is the famous Shunkov theorem on the local finiteness of a periodic group $G$ having an involution whose centralizer in $G$ is finite.