I'm concerned about the group structure on $[X,S^n]$, i.e. the set of homotopy classes of continuous maps from $X$ to $S^n$.

On the one hand, $[X,Y]$ has a group structure that is natural with respect to $X$ if and only if $Y$ is an H-space. The naturality is in the sense that $f:X'\to X$ induces a homomorphism $f_{\star}:[X,Y]\to[X',Y]$. It's known that $S^n$ is an H-space only for $n=0,1,3,7$. Thus, for a general $X$, there's no natural group structure on $[X,S^n]$ when $n$ takes other values.

On the other hand, when $X$ is a closed smooth $d$-manifold, the Pontryagin-Thom construction establishes a bijection between $[X,S^n]$ and $\Omega_{d-n}^{fr}(X)$, i.e. the (unstable) $(d-n)$th framed bordism set of $X$. When $d<2n-1$, two transverse $(d-n)$-submanifolds in $X$ have no intersection and the disjoint union induces a group structure on $\Omega_{d-n}^{fr}(X)$. Then we can make $[X,S^n]$ into a group via the Pontryagin-Thom bijection.

Hence $[X,S^n]$ is a group when $X$ is a closed smooth manifold whose dimension $<2n-1$. I would like to know how "natural" or how "unnatural" this group structure is: (1) if $X'$ is a closed smooth manifold whose dimension $<2n-1$, does a smooth map $f:X'\to X$ induce a group homomorphism? (2) When $n=0,1,3,7$, does this group structure coincide with the H-space induced group structure?

**Edit 1** The question has been edited to correct a mistake pointed out by Gregory.

**Edit 2** As mentioned by Tyrone, $[X,S^n]$ is a cohomotopy group when $X$ is a complex of dimension $<2n-1$, with the multiplication induced by the folding map. It reduces to the H-space induced group if $n=0,1,3,7$. My **Q2** essentially asks whether *cohomotopy groups* coincide with *framed bordism groups* (when the latter applies).