  Let $f $ be a Morse function on a smooth compact surface $M$ and
$\mathcal{S}'(f)$ be a group of $f$-preserving diffeomorphisms of $M$ which are
isotopic to the identity map. Let also $G(f)$ be a group of automorphisms of
the graph of $f$ induced by elements from $\mathcal{S}'(f)$, and $\Delta'$ be a
subgroup of $\mathcal{S}'(f)$ of diffeomorphisms which trivially act on the
graph of $f$ and are isotopic to the identity map. The group
$\pi_0\mathcal{S}'(f)$ can be viewed as an analogue of a mapping class group
for $f$-preserved diffeomorphisms of $M$. Groups $\pi_0\Delta'(f)$ and $G(f)$
can be viewed as groups which encode `combinatorially trivial' and
`combinatorially nontrivial' counterparts of $\pi_0\mathcal{S}'(f)$
respectively. In the paper we compute groups $\pi_0\mathcal{S}'(f)$, $G(f)$,
and $\pi_0\Delta'(f)$ for Morse functions on $2$-torus $T^2$.
