Let $G$ be an undirected Cayley graph over an abelian group. Let $H$ a regular graph whose independence number and chromatic number are known. Let $inj(G,H)$ be the number of injective homomorphisms from $G$ to $H$. It is known SUBGRAPH ISOMORPHISM is NP-complete. Consider the CAYLEY SUBGRAPH ISOMORPHISM problem:
Given an undirected Cayley $G$ and a regular $H$, is inj(G,H)>0?
Is the above problem NP-complete?
Since linear codes are abelian, I am extending the question. Given two $[n_i,k_i,d_i]$ linear codes $C_i$ for $i=1$ and $2$ and $n_1<n_2$, is deciding $C_1\cong D \subset C_2$ NP-complete? Note that code isomrphism is similar to graph isomorphism.