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The Regular Post Embedding Problem is a variant of Post's Correspondence Problem where one compares strings with the subword relation and imposes additional regular constraints on admissible solutions. It is known that this problem is decidable, albeit with very high complexity. We consider and solve variant problems where the set of solutions is compared to regular constraint sets and where one counts the number of solutions. Our positive results rely on two non-trivial pumping lemmas for Post-embedding languages and their complements.