Question:Isomorphic normal subgroups isomorphic quotient groups equivalent as subgroups

Q: '''Suppose $$H$$ and $$K$$ are normal subgroups of $$G$$ such that $$H$$ and $$K$$ are isomorphic groups and the quotient groups $$G/H$$ and $$G/K$$ are isomorphic groups. Is there an automorphism of $$G$$ sending $$H$$ to $$K$$?'''

A: Not in general, though it is hard to construct counterexamples accessible at a very elementary level. A discussion of many minimal counterexamples (with various additional restrictions on the nature of the groups) is available at answer references::series-equivalent not implies automorphic. The jargon used there is: two subgroups are termed answer references::series-equivalent subgroups if they are isomorphic normal subgroups and the quotient groups are isomorphic. Two subgroups are termed answer references::automorphic subgroups if there is an automorphism of the group mapping one to the other.

More basic: Question:Normal subgroup quotient group determine whole group