Compute sum over bounded numbers prime with given number

Question

When I was doing some task of analytic number theory I was stuck on computing this sum $$S:=\frac{1}{L} \sum_{q \in \mathcal{Q}} \phi(q) \overline{a}^{\frac{1}{2}},$$ where $\overline{a}$ is the inverse of $a$ modulo $q>0,$ $\mathcal{Q}$ is a nonempty set of numbers defined by $$\mathcal{Q}=\left\{q \in [Q,2Q]; \ \gcd(a,q)=1\right\}, \quad Q\geq 1$$ and $$L=\sum_{q \in \mathcal{Q}} \phi(q),$$ $\phi$ is the Euler function. Can someone help me in doing this task? Any help is appreciated.