This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received no answers/comments so far, so here I'm posting a more specific question with some commentary, highlighting changes from the linked question.
Let $p:E \rightarrow B$ be a fibration where E and B are path-connected, locally path-connected and compact manifolds. Let $\Sigma_{n}$ denote the symmetric group on $n$ symbols, and let $\Sigma_{n}$ act on $E$ (non-freely). Let us have an induced map $f:E/\Sigma_{n} \to B$ such that $p = fq$, where $q:E \to E/\Sigma_{n}$ is the quotient map. Can we now say that $f:E/\Sigma_{n} \to B$ is a Serre fibration? Can a cube $I^{k}$ in orbit space $E/\Sigma_{n}$ be lifted to some cube $I^{m_{k}}$ in $E$ for some $m_{k} \leq k$? Maybe the slice theorem can be of use here but I'm not certain.
