Map from Torus to Projective Plane inducing an Epimorphism on Fundamental groups

Question

Is it true or false that there exists a continuous map from the torus to the projective plane inducing an epimorphism on the fundamental groups?

I am quite lost on this problem, what I know is that $\pi_1(T)=\mathbb{Z}\times\mathbb{Z}$, that $\pi_1(\mathbb{R}P^2)=\mathbb{Z}/2\mathbb{Z}$, and that an epimorphism is a surjective homomorphism.

Let $f:T\to \mathbb{R}P^2$, I tried $f_*:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}$, $f_*(x,y)=xy\pmod 2$ which is certainly surjective, but I am skeptical it fits the requirements of this question.

Answer 1

From the comment:

$(1)$ Find a continuous map $f : \mathbb S^1 \times \mathbb S^1 \to \mathbb S^1$ so that $f_*$ is surjective.

$(2)$ Let $\gamma : \mathbb S^1 \to \mathbb{RP}^2$ represent the generator of $\pi_1(\mathbb{RP}^2)$. Then $\gamma_*$ is surjective.

$(3)$ The composition $\gamma \circ f$ is one of the example as $(\gamma\circ f)_* = \gamma_* \circ f_*$ is surjective.