Let's imagine I've got $Q_1,\ldots,Q_L$ independent identically distributed normal random variables with parameters $\mu, \sigma^2$, and $L$ is binomial with parameters $n,p$.
Let $Y=Q_1+\cdots+Q_L$.
Then $M_Y(s)$ is a random variable satisfying $$M_Y(s \mid L=l) = e^{l\left( \mu s + \frac{1}{2} \sigma^2 s^2 \right)}.$$
Is the moment generating function for $Y$ equal to:
$$\begin{align} M_Y(s) &= \sum_k {n \choose k} p^k(1-p)^{n-k}M_Q(s)^k\\ &= \sum_k {n \choose k} \left( p M_Q(s) \right)^k (1-p)^{n-k}\\ &= \cdots\text{ ?} \end{align}$$
Hint: \begin{align} \sum_k {n \choose k} \left( p M_Q(s) \right)^k (1-p)^{n-k}= \left(p M_Q(s)+1-p\right)^L \end{align}