Groups containing a strongly embedded subgroup

An involution $v$ of a group $G$ is said to be finite (in $G$) if $vv^g$ has finite order for any $v\in G$. A subgroup $B$ of $G$ is called a strongly embedded (in $G$) subgroup if $B$ and $G\setminus B$ contain involutions, but $B\cap B^g$ does not, for any $g\in G\setminus B$.
We prove the following results.
Theorem 1. Let a group $G$ contain a finite involution and an involution whose centralizer in $G$ is periodic. If every finite subgroup of $G$ of even order is contained in a simple subgroup isomorphic, for some $m$, to $L_2(2^m)$ or $Sz(2^m)$, then $G$ is isomorphic to $L_2(Q)$ or $Sz(Q)$ for some locally finite field $Q$ of characteristic two. In particular, $G$ is locally finite.
Theorem 2. Let a group $G$ contain a finite involution and a strongly embedded subgroup. If the centralizer of some involution in $G$ is a 2-group, and every finite subgroup of even order in $G$ is contained in a finite non-Abelian simple subgroup of $G$, then $G$ is isomorphic to $L_2(Q)$ or $Sz(Q)$ for some locally finite field $Q$ of characteristic two.