Normal closure-finding problem

Description
The normal closure-finding problem is the general name for a class of problems where a group $$G$$ is described using a group description rule, a subgroup $$H$$ is described using a subgroup description rule, and our goal is to use a (specified) subgroup description rule to find the normal closure $$H^G$$.

Problems it reduces to

 * Normality testing problem: See black-box reduction of normal closure-finding in terms of generating sets to normality testing.