Additive maps on C$^*$-algebras commuting with $|.|^k$ on normal‎ ‎elements

‎Let $\mathcal {A} $ and $\mathcal {B} $ be C$^*$-algebras‎. ‎Assume‎ ‎that $\mathcal {A}$ is of real rank zero and unital with unit $I$‎ ‎and $k > 0$ is a real number‎. ‎It is shown that if $\Phi:\mathcal{A}‎ ‎\to\mathcal{B}$ is an additive map preserving $|\cdot|^k$ for all‎ ‎normal elements; that is‎, ‎$\Phi(|A|^k)=|\Phi(A)|^k $ for all normal‎ ‎elements $A\in\mathcal A$‎, ‎$\Phi(I)$ is a projection‎, ‎and there‎ ‎exists a positive number $c$ such that $\Phi(iI)\Phi(iI)^{*}\leq‎ ‎c\Phi(I)\Phi(I)^{*}$‎, ‎then $\Phi$ is the sum of a linear Jordan‎ ‎*-homomorphism and a conjugate-linear Jordan‎ ‎*-homomorphism‎. ‎If‎, ‎moreover‎, ‎the map $\Phi$ commutes with‎ ‎$|.|^k$ on $\mathcal{A}$‎, ‎then $\Phi$ is the sum of a linear‎ ‎*-homomorphism and a conjugate-linear *-homomorphism‎. ‎In the case when $k \not=1$‎, ‎the assumption $\Phi(I)$ being a projection can be‎ ‎deleted‎.