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Post Closed as "Not suitable for this site" by Derek Holt, user6976, YCor, Stefan Kohl♦, Neil Strickland
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Let $G$ be a finite group such that $Z(G)\leq SL_2(q) \leq G $ and $G/Z(G) \cong PGL_2(q)$· Is $G$ a subgroupthere any information about the structure of $ GL_2(q)$ $G$?

Let $G$ be a finite group such that $Z(G)\leq SL_2(q) \leq G $ and $G/Z(G) \cong PGL_2(q)$· Is $G$ a subgroup of $ GL_2(q)$ ?

Let $G$ be a finite group such that $Z(G)\leq SL_2(q) \leq G $ and $G/Z(G) \cong PGL_2(q)$· Is there any information about the structure of $G$?

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sara
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Let $G$ be a finite group such that $Z(G)\leq SL_2(q) \leq G $ and $G/Z(G) \cong PGL_2(q)$· Is $G\cong GL_2(q)$ the only possibilty for $G$ a subgroup of $ GL_2(q)$ ?

Let $G$ be a finite group such that $Z(G)\leq SL_2(q) \leq G $ and $G/Z(G) \cong PGL_2(q)$· Is $G\cong GL_2(q)$ the only possibilty for $G$?

Let $G$ be a finite group such that $Z(G)\leq SL_2(q) \leq G $ and $G/Z(G) \cong PGL_2(q)$· Is $G$ a subgroup of $ GL_2(q)$ ?

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The finite extensions of $SL_2(q)$

Let $G$ be a finite group such that $Z(G)\leq SL_2(q) \leq G $ and $G/Z(G) \cong PGL_2(q)$· Is $G\cong GL_2(q)$ the only possibilty for $G$?