We study finite ontology mediated query answering (FOMQA), the variant of ontology mediated query answering (OMQA) where the represented world is assumed to be finite, and thus only finite models of the ontology are considered. The query language we study is conjunctive two-way regular path queries (C2RPQ), which can be regarded as the result of adding simple recursion to Conjunctive Queries. We focus on understanding the finite-controllable fragments of C2RPQ, that is, on the question: for which fragments of C2RPQ the OMQA and FOMQA are equivalent? For graph classes S, we consider fragments C2RPQ(S) of C2RPQ as the queries whose underlying graph structure is in S. We completely classify the finite-controllable and non-finite-controllable fragments under: inclusion dependencies, (frontier-)guarded rules, frontier-one rules (either with or without constants), and more generally under guarded-negation first-order constraints. For the finite-controllable fragments, we show a reduction to the satisfiability problem for guarded-negation first-order logic, yielding a 2EXPTIME algorithm (in combined complexity) for the corresponding (F)OMQA problem.