  Let $\mathbb{H}^{n}$ be the Heisenberg group. For $0 \leq \alpha < Q=2n+2$
and $N \in \mathbb{N}$ we consider exponent functions $p(\cdot) :
\mathbb{H}^{n} \to (0, +\infty)$, which satisfies H\"older conditions, such
that $\frac{Q}{Q+N} < p_{-} \leq p(\cdot) \leq p_{+} < \frac{Q}{\alpha}$. In
this article we prove the $H^{p(\cdot)}(\mathbb{H}^{n}) \to
L^{q(\cdot)}(\mathbb{H}^{n})$ and $H^{p(\cdot)}(\mathbb{H}^{n}) \to
H^{q(\cdot)}(\mathbb{H}^{n})$ boundedness of convolution operators with kernels
of type $(\alpha, N)$ on $\mathbb{H}^{n}$, where $\frac{1}{q(\cdot)} =
\frac{1}{p(\cdot)} - \frac{\alpha}{Q}$. In particular, the Riesz potential on
$\mathbb{H}^{n}$ satisfies such estimates.
