$E^2$-term $$E^2_{pq}=H^{-p}_G(X,A;\pi _q ({\mathcal F}))$$ for $p+q\geq 0$ and $E^2_{pq}=0$ otherwise, converging to $\pi _{p+q}(F_u(X,A;E,B)^G,u)$ when $p+q>0$.

The spectral sequence above is called the equivariant Federer spectral sequence since it is the equivariant version of a spectral sequence introduced by Federer [@federer]. We will be using this spectral sequence for the following special case: Let $X$ be a finite $G$-CW-complex such that $X^H$ is a path connected simple space for all $H \leq G$. Take $A=\emptyset $, $E=X$, $B=*$, $p:E\to B$ to be the constant map, and $u:X\to E$ to be the identity map. Then $F_{{\mathrm{id}}} (X, \emptyset ; X , * )$ will be homotopy equivalent to the identity component of $\operatorname{Aut}_G(X)$. Since all the components of $\operatorname{Aut}_G(X)$ have the same homotopy type, we have $$\pi _n (\operatorname{Aut}_G(X)){\cong}\pi _n (F_{{\mathrm{id}}} (A, \emptyset; X, *))$$ for all $n>0$. So, we can use the equivariant Federer spectral sequence to calculate the homotopy groups of $\operatorname{Aut}_G (X)$.

Also note that in the situation we consider, the local coefficient system is constant on orbits. Bredon cohomology with coefficients in a $G$-equivariant local coefficients system has an alternative description when the coefficient system is constant on $G$-orbits. This description involves the modules over the 