An alternative approach would be to use cumulants. The fourth central moment of a random variable $X$ can be expressed in terms of cumulants as follows: $\mu_4(X)=\kappa_4(X)+3\kappa^2_2(X).$
Now, cumulants add over independent random variables and the second cumulant is just the variance, i.e., $\kappa_2=\mu_2$.
Writing $Y=\sum_{i=1}^n Z_i$, where the $Z_i\,$s are i.i.d. random variables, we have
\begin{eqnarray*} \mu_4(Y)&=&\kappa_4(Y)+3\kappa^2_2(Y)\\ &=&n\kappa_4(Z)+3[n\kappa_2(Z)]^2\\ &=&n\left[\mu_4(Z)-3\kappa_2^2(Z)\right]+3[n\kappa_2(Z)]^2\\ &=&n\, \mu_4(Z) +3n(n-1)\,\mu_2^2(Z). \end{eqnarray*}