For the model two-complex K of the group presentation ⟨x,y | x^k+1yxy⟩, with odd k greater than or equal to 1, we describe representatives for all free and based homotopy classes of maps from K into the projective plane. As a result we classify the homotopy classes containing only surjective maps. With this approach we get an answer, for maps into the real projective plane, to a classical question in topological root theory, which is known so far, in dimension two, only for maps into the sphere, the torus and the Klein bottle. The answer follows by proving that for all odd k greater than or equal to 1, the two-complex K has trivial second integer cohomology group and, for odd k greater than or equal to 3, there exist strongly surjective maps from K onto the real projective plane. For k=1, there does not exist such a strongly surjective map.

Part of this work was developed during the visit of the second author to the Faculdade de Matemática -- Universidade Federal de Uberlandia, Uberlandia MG, during the period Oct 31 - Nov 03, 2019. The second author would like thank the Faculdade de Matem´tica for the great hospitality. This work is part of the Projeto Temático FAPESP: Topologia Algébrica, Geométrica e Diferencial -- 2016/24707-4 (Brazil) and Projeto DIRPE/PSFE No. 0022/2020 -- UFU. We would like to thank the referee for his/her valuable suggestions improving the presentation of the early version of the manuscript.