Group satisfying Oliver's condition

Definition
A group satisfying Oliver's condition is a finite group $$G$$ satisfying the following equivalent conditions:


 * $$G$$ is a $$q$$-group extension of a cyclic extension of a $$p$$-group: There exist primes $$p,q$$ (possibly equal) and subgroups $$N \le H \le G$$, such that the following hold. $$N$$ is a normal subgroup of $$H$$, $$H$$ is a normal subgroup of $$G$$, $$G/H$$ is a $$q$$-group, $$H/N$$ is a cyclic group, and $$N$$ is a $$p$$-group. By a $$p$$-group (resp., $$q$$-group), we mean a group of prime power order where the underlying prime is $$p$$ (resp., $$q$$).
 * Any action of $$G$$ on a contractible finite simplicial complex has a fixed face (in the geometric realization, a fixed point).