I'm trying to solve the following problem that can be found in Kosniowski's a first course in algebraic topology. This problem is under the chapter on Borsuk Ulam theorem problem $20.7$ d.
Question
Suppose that $X$ and $Y$ are path connected $G$-spaces for which the action of G is properly discontinuous. Let $p:X\to X/G$ and $q:Y\to Y/G$ denote the canonical projections. Suppose also that $\varphi:X\to Y$ is a $G$ equivariant continuous map. Let $\phi: X/G\to Y/G$ denote the map induced by $\varphi$. Prove that the homomorphism: $$\phi_*: \pi(X/G,p(x_0))\to \pi(Y/G,q\varphi(x_0))$$ induces a isomorphism: $$\pi(X/G,p(x_0))/p_*(X,x_0)\to \pi(Y/G,q\varphi(x_0))/q_*\pi(Y,\varphi(x_0))$$
My attempt
I've been trying to solve the problem under the assumption that the paths X,Y are path connected, the original problem doesn't give this hypothesis, but I believe it's a mistake. Given those conditions we would have that $p,q$ are coverings of $X/G,Y/G$ respectively. Then there's a theorem that says that the groups $\pi(X/G,p x_0)/p_*\pi(X,x_0)$ and $G$ are isomorphic. So we would have have that the groups $$\pi(X/G,p(x_0))/p_*(X,x_0)\cong G\cong \pi(Y/G,q\varphi(x_0))/q_*\pi(Y,\varphi(x_0))$$
And so in fact they are isomorphic, this however doesn't prove that the homomorphism induced by $\phi_*$ is an isomorphism.
Any help on how to solve this problem would be highly appreciated.