Puppe sequence is exact after mapping into a based space

At $[Y,Z{]}_{\ast }$, if $g:Y\to Z$ extends to $G:C_f\to Z$, the cone formula $G([x,t])$ is a based nullhomotopy of gf. Conversely a based nullhomotopy $H(x,t)$ of gf is constant on the cone tip and basepoint track, so it defines $CX\to Z$. It agrees with g at the attaching base. F2 pastes them to $G:C_f\to Z$ with G|Y=g. Thus the image of restriction is exactly the distinguished fibre of precomposition with f.

The same argument applies to every map h followed by its cofiber inclusion. F3 identifies each consecutive pair in the iterated cofiber sequence, up to based homotopy equivalences and its specified reflection, with such a pair. Precomposition by a based homotopy equivalence has inverse on mapping classes given by its homotopy inverse, since composing either inverse homotopy with a map preserves its basepoint. Transporting step 1.1 across those bijections proves exactness at every displayed eligible term.

F4 gives the group and abelian ranges. Precomposition by an unreflected suspension is a homomorphism. Parameter reflection sends every class to its inverse, by the reversal homotopy in F4, so a reflected arrow is an antihomomorphism: $T(ab)=T(b)T(a)$. On abelian groups it is a homomorphism. Removing a reflection does not change the distinguished fibre, because inversion fixes only the identity over the identity; it does not change the image, because the image of the unreflected homomorphism is a subgroup and is closed under inverses. Thus removing all reflection signs preserves each kernel and image, yielding the asserted exact sequence of groups on the suspended portion. No nonabelian inversion map is claimed to be a homomorphism.