Groups Containing a Self-Centralizing Subgroup of Order 3

In 1962 Feit and Thompson obtained a description of finite groups containing a subgroup $X$ of order 3 which coincides with its centralizer. This result is carried over arbitrary groups with the condition that $X$ with every one of its conjugates generate a finite subgroup. We prove the following theorem.
Theorem. Suppose that a group $G$ contains a subgroup $X$ of order $3$ such that $C_G(X)=\langle X\rangle$. If, for every $g\in G$, the subgroup $\langle X,X^g\rangle$ is finite, then one of the following statements holds:
$(1)$ $G=NN_G(X)$ for a periodic nilpotent subgroup $N$ of class $2$, and $NX$ is a Frobenius group with core $N$ and complement $X$.
$(2)$ $G=NA$, where $A$ is isomorphic to $A_5\simeq SL_2(4)$ and $N$ is a normal elementary Abelian $2$-subgroup; here, $N$ is a direct product of order $16$ subgroups normal in $G$ and isomorphic to the natural $SL_2(4)$-module of dimension $2$ over a field of order $4$.
$(3)$ $G$ is isomorphic to $L_2(7)$.
In particular, $G$ is locally finite.