  The spherical average $A_{1}(f)$ and its iteration $(A_{1})^{N}$ are
important operators in harmonic analysis and probability theory. Also $\Delta
(A_{1})^{N}$ is used to study the $K$ functional in approximation theory, where
$\Delta $ is the Laplace operator. In this paper, we obtain the sufficient and
necessary conditions to ensure the boundedness of $\Delta (A_{1})^{N}$ from the
modulation space $M_{p_{1},q_{1}}^{s_{1}}$ to the modulation space
$M_{p_{2},q_{2}}^{s_{2}}$ for $1\leq p_{1},p_{2},q_{1},q_{2}\leq \infty $ and
$s_{1},s_{2}\in \mathbb{R}$.
