In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph $G$ has a symmetry of prime order $p$, then its characteristic polynomial, with coefficients in the finite filed $\mathbb{F}_p$, is determined by the characteristic polynomial of its quotient graph $\overline G$. Similar results are also proved for some generalization of the Tutte polynomial.