Cover for a group

Definition
A family $$\mathcal{F}$$ of subgroups of a group $$G$$ is termed a cover for $$G$$ if it satisfies the following conditions:


 * If $$H \in \mathcal{F}$$, and $$x \in G$$, then $$H^x \in \mathcal{F}$$
 * if $$M$$ is a maximal subgroup of $$G$$ and $$y \in M$$ has prime power order, then there exists $$H \in \mathcal{F}$$ such that $$y \in H$$ and $$H \cap M \triangleleft H$$

Inductive cover
A cover of a group is said to be inductive if given any subnormal subgroup of the group, intersecting each member of the cover with that subnormal subgroup gives a cover of that subgroup.