Abstract: Let W be a twisted I-bundle over a Klein bottle fibered trivially and V be a type-(p,q) exceptionally fibered solid torus. Since partial V and partial W is a torus, we may form a quotient space N by identifying the boundaries via a fiber preserving homeomorphism. Then, N is a Seifert fiber space with one exceptional fiber of type-(p,q) whose base space is a projective plane containing the exceptional point of order p In the talk, we see that N is double covered by a symmetric lens space L which gives a geometric structure on N This allows us to compute the isometry group of N denoted by Isom(N) It turns out that L and Isom(N) are completely determined by the initial assignment of a type-(p,q) fibration on V In other words, we will classify how a fibration type affects topological structures of N and L as well as Isom(N)