Post's Correspondence Problem (the PCP) is a classical decision problem in theoretical computer science that asks whether for pairs of free monoid morphisms g,h:Σ∗→Δ∗$g, h\colon \Sigma ^*\rightarrow \Delta ^*$ there exists any non‐trivial x∈Σ∗$x\in \Sigma ^*$ such that g(x)=h(x)$g(x)=h(x)$. PCP for a group Γ$\Gamma$ takes pairs of group homomorphisms g,h:F(Σ)→Γ$g, h\colon F(\Sigma)\rightarrow \Gamma$ instead, and similarly asks whether there exists an x$x$ such that g(x)=h(x)$g(x)=h(x)$ holds for non‐elementary reasons. The restrictions imposed on x$x$ in order to get non‐elementary solutions lead to several interpretations of the problem; we mainly consider the natural restriction asking that x∉ker(g)∩ker(h)$x \notin \ker (g) \cap \ker (h)$ and prove that the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic Γ$\Gamma$, but decidable when Γ$\Gamma$ is virtually nilpotent, en route also studying this problem for finite extensions. We also consider a different interpretation of the PCP due to Myasnikov, Nikolaev and Ushakov, proving decidability for torsion‐free nilpotent groups. [ABSTRACT FROM AUTHOR]