Unbased homotopic based maps need not induce the same based homotopy map without basepoint transport

Let W be the wedge of two quotient circles, with vertex w and standard loops a,b. It has one vertex and two attached 1-cells, hence is a finite CW complex. By F1 the loop a:w×I→W extends to a homotopy $H:W\times I\to W$ with H(-,0)=id. Set g=H(-,1). Since a(0)=a(1)=w, both id and g are based; H is a free homotopy between them with basepoint track a.

F2 gives ${\mathrm{i}\mathrm{d}}_{\ast }={\beta }_a{g}_{\ast }$ and ${\beta }_a(c)=ac{a}^{-1}$. Applying it to b and multiplying in the free group yields $g_{\ast }(b)=a^{-1}ba$. By F3 and F4 this is the reduced three-letter word a inverse, b, a; it has no adjacent cancellation and differs from the one-letter reduced word b. Therefore $g_{\ast }\mathrm{\ne }{\mathrm{i}\mathrm{d}}_{\ast }$, although the maps are freely homotopic.