Given a Galois cover of curves $f$ over a field of characteristic $p$, the lifting problem asks whether there exists a Galois cover over a complete mixed characteristic discrete valuation ring whose reduction is $f$ . In this thesis, we try to answer this question in the case where the Galois groups are elementary abelian $p$-groups. We prove a combinatorial criterion for lifting an elementary abelian $p$-cover, dependent on the branch loci of its $p$-cyclic subcovers. Moreover, we study how branch points of a lift coalesce on the special fiber. Finally, we construct lifts for several families of $(\mathbb{Z}/2)^3$-covers of various conductor types, both with equidistant branch locus geometry and non-equidistant branch locus geometry.