I hope I've understood your question.
Any connected based space $X$ is homotopy equivalent to the classifying space of a topological group $G$ in a functorial way (the group is the realization of the Kan loop group of a the simplicial total singular complex). So we can assume $X = BG$. Then there is an equivalence of homotopy categories $$ \text{ho(parametrized spectra}/X) \qquad \simeq \quad \text{ho(naive $G$-spectra)} $$ (This equivalence arises from a Quillen equivalence with respect to suitable choices of model structures.)
The functors inducing the equivalences are given from right to left by taking the unreduced Borel construction and from left to right by base change along $EG \to X$ and then collapsing out the zero-section.
Actually the situation above arises from a Quillen equivalence between the model categories "Spaces over X" and and "$G$-spaces." Then one can pass to categories spectra on both sides to get the result.
