For more details see the following (and the references therein):

• Homomorphisms and oriented colorings of equivalence classes of oriented graphs, W. Klostermeyer and G. MacGillivray, Disc. Math, 2004
• Pushing Vertices and Orienting Edges; W. Klostermeyer, Ars Combinatoria 1999.
  See Errata
• An Extremal Connectivity Parameter of Tournaments; W. Klostermeyer and L. Soltes, Discrete Mathematics 2000
• An Analog of Camion's Theorem in Squares of Cycles; W. Klostermeyer, Congressus Numerantium 1998
• Hamiltonicity and Reversing Arcs in Digraphs ; W. Klostermeyer and L. Soltes, Journal of Graph Theory, 1998

Other assorted facts:
1. Any tournament with at least 7 vertices can be made strong by pushing at most one vertex (and there are at least two such vertices).
2. Given a tournament with at least 7 vertices. If as many as n-6 vertices are marked as "forbidden", you can push unforbidden vertices so as to make to make the tournament strong. (To see this, first note that the forbidden sub-tournament has a Hamiltonian path and focus on the two end vertices of that path).
3. All tournaments with 11 vertices can be made 2-connected using pushes. Thus the only open case is for 10 vertices.