Let $\mathcal{F}$ be a locally constant étale sheaf of modules on a scheme $X$. If $\phi:X\rightarrow Y$ is étale and surjective, can I say that $\phi_*\mathcal{F}$ is locally constant?
My idea is that if $\{U_i\rightarrow X\}$ is an étale cover of $X$ such that $\mathcal{F}|_{U_i}$ is constant, then $\{U_i\rightarrow X\rightarrow Y\}$ should be a cover of $Y$ such that $\phi_*\mathcal{F}|_{U_i}$ is constant.
How far is this from being true? What condition could I put on $\phi$ to assure that $\phi_*\mathcal{F}$ is locally constant?