There are many non-homeomorphic 3-manifolds with isomorphic fundamental groups, for example the lens spaces $L(p,q_1)$ and $L(p,q_2)$ with $q_1 \ne \pm q_2^{\pm 1}\mod p$. Also, Seifert fibered spaces over the disk with two exceptional fibers can have isomorphic fundamental groups. A third type of example can be constructed by using the connect sum of knot of a chiral knot $K$, $S^3-(K\#K)$ and $S^3-(K\#-K)$ (think of the square and granny knots). Even more examples can be constructed by connect summing the examples in this paragraph (in the sense of 3-manifolds as opposed to knots).

However, there are also non-orientable and orientable manifolds with isomorphic fundamental groups, which motivates my question:

Is there a list of all non-homeomorphic pairs of 3-manifolds (orientable and non-orientable) with isomorphic fundamental groups?

Given, the likely complexities of reducible manifolds and graph manifolds, it would be fine if it was only a list of geometric 3-manifolds.