= Solution
View the <torus> as the topological <group> $S^1\times S^1$ under coordinatewise multiplication, and let $x_0$ be the common basepoint. From the given free <homotopy> construct
$$
\boxed{K(s,z)=H(s,z)\,H(s,1)^{-1}x_0}.
$$
This is continuous, has $K(s,1)=x_0$, and at $s=0,1$ agrees with $\alpha,\beta$ because $H(0,1)=H(1,1)=x_0$. Thus it exhibits the requested <based homotopy> explicitly.
Algebraically $\pi_1(S^1\times S^1)\cong\mathbb Z^2$ is <Abelian>, so conjugate elements are equal. Free and <based homotopy> classes of loops with a specified common basepoint therefore coincide here.
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