So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and solvable extensions unramified everywhere for his students to play with and that he had find this easy to construct with class field theory in the back of his head. But then he asked me if I knew how to construct an extension of number fields with Galois group $A_{5}$ and unramified everywhere. All I could say at the time (and now) is:

1. There are Hilbert modular forms unramified everywhere.
2. There are Hilbert modular forms whose residual $G_{{F}_{v}}$-representation mod $p$ is trivial for all $v|p$.
3. There are Hilbert modular forms whose residual $G_{F}$-representation mod $p$ has image $A_{5}$ inside $\operatorname{GL}_{2}(\mathbb F_{p})$.

Suppose there is a Hilbert modular form satisfying all three conditions. Then the Galois extension through which its residual $G_{F}$-representation factors would have Galois group $A_{5}$ and would be unramified everywhere.

Can this be made to work?

Regardless of the validity of this circle of idea, can you construct an extension of number fields unramified everywhere and with Galois group $A_{5}$?