= Solution
Chain maps $f,g:C_*\to C_*'$ are <chain homotopy>[chain homotopic] when maps $h_n:C_n\to C'_{n+1}$ satisfy
$$
f-g=\partial h+h\partial.
$$
For a cycle $z$, $f(z)-g(z)=\partial h(z)$ is a boundary, so $f_*=g_*$ on homology.
Fix a vertex $e_0$ of $\Delta^n$ and define the cone operator by adjoining $e_0$ to an oriented simplex, with zero when it is already present. The simplicial boundary formula gives
$$
\partial h+h\partial=\operatorname{id}
$$
on the reduced chain complex. Thus its identity is null-homotopic and
$$
\widetilde H_k(\Delta^n)=0
$$
for every $k$; equivalently, $H_0(\Delta^n)\cong\mathbb Z$ and higher homology vanishes.
For the 2-skeleton $K$ of $\Delta^6$,
$$
\dim C_0=7,
\quad\dim C_1=21,
\quad\dim C_2=35.
$$
It is connected and simply connected, so $H_0\cong\mathbb Z$ and $H_1=0$. Euler characteristic then gives
$$
21=1+\operatorname{rank}H_2,
\qquad
\boxed{H_2(K)\cong\mathbb Z^{20}}.
$$
For $\sigma=(0123)(456)$, no vertex or edge is fixed. The only invariant 2-simplex is $[456]$, on which the 3-cycle preserves orientation, so the chain traces are $0,0,1$. The <Lefschetz number>[Lefschetz trace formula] gives
$$
1=\operatorname{tr}(f_*|H_0)+\operatorname{tr}(f_*|H_2),
$$
therefore
$$
\boxed{\operatorname{tr}(f_*|H_2(K;\mathbb Q))=0}.
$$
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