If $M$ is connected, then @MarkGrant's fibration sequence gives a long exact sequence on homotopy groups showing that $\pi_1(M/\Sigma_k)\to \Sigma_k$ is surjective. Now apply the $K(-,1)$-functor and obtain a map $K(\pi_1(M)/\Sigma_k)\to B\Sigma_k$. The corresponding $\Sigma_k$-bundle over $K(\pi_1(M)/\Sigma_k)$ has as total space $K(\pi_1(M))$, so there exists a model of your map $F$ which is a covering map. You have defined $F$ only up to homotopy, so I think your question needs to be whether one can find a model of $F$ which is a covering map.
user83633
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