Motivated by these posts, An NP-complete variant of factoring and Relationship between symmetry and computational intractability, It seems to be worthwhile to investigate the different factors that increase the hardness of problems in NPI. I'm interested in other NP-complete variants of problems of intermediate complexity.
Is there a recent survey paper of $NP$-complete variants of problems of intermediate complexity (problems between P and NP-complete)?
Cross posted on TCS StackExchange
UPDATE: Bounty may be awarded to an answer that gives other interesting NPI problems that have an $NP$-complete variant.