Invariant Heegaard Surfaces in Manifolds with Involutions and the   Heegaard Genus of Double Covers
  Let M be a 3-manifold admitting a strongly irreducible Heegaard surface S and f:M \to M an involution. We construct an invariant Heegaard surface for M of genus at most 8 g(S) - 7. As a consequence, given a (possibly branched) double cover \pi:M \to N we obtain the following bound on the Heegaard genus of N: g(N) \leq 4g(S) - 3. We also get a bound on the complexity of the branch set in terms of g(S). If we assume that M is non-Haken, by Casson and Gordon we may replace g(S) by g(M) in all the statements above. 
Yo'av Rieck and J Hyam Rubinstein