On centralizers of involutions in simple groups

In this paper we prove the following
Theorem. Let $G$ be a inite simple group$,$ $t$ an involution of $G$ and $C(t)$ the centralizer of $t$ in $G$. If $L(C(t))\simeq\langle t\rangle\times PSL(2,q)$ where $q<3,$ then a Sylow $2$-subgroup of $G$ is an elementary group of order $8$.
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