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Given a topological group $G$ acting on topological space $X$, transitively by $a:G\times X\rightarrow X$ and I have a sheaf $F$ of finite dimensional spaces, such that $a^*F\simeq \pi^*F$ with $\pi:G\times X\rightarrow X$ the projections, what can I say about $F$? (besides the stalks being isomorphic)

Is it true that $F$ is a constant sheaf?

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    And the sheaf $F$ is constructible. – 2017-03-10

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