A full subcomplex makes $\\Omega\\mathcal Z_K$ a retract of $\\Omega\\mathcal Z_L$: loop-homology embedding in every degree

Let $K$ be a finite abstract simplicial complex on $s$ vertices, $L$ one on $m$ vertices, and let $e:V(K)\hookrightarrow V(L)$ identify $K$ with the full subcomplex of $L$ spanned by $e(V(K))$, i.e. $\sigma$ is a face of $K$ if and only if $e(\sigma)$ is a face of $L$. Then for every degree $q$ there is an injective additive homomorphism

The coordinate inclusion $\mathcal Z_K\to\mathcal Z_L$ (extend by $1$ outside $e(V(K))$) and the coordinate projection $\mathcal Z_L\to\mathcal Z_K$ are both based maps for the all-ones basepoints, and the projection is a left inverse of the inclusion. Post-composition with a based map is continuous for the compact-open topology on loop spaces, so the based loop spaces inherit the retraction, and any homology functor converts it into a split injection.

Formalization Note The loop space is the space of based paths from the all-ones point to itself with the compact-open topology, homology is integral singular homology in the fixed degree $q$, and the conclusion is an explicit injective additive homomorphism.