Give an example of a continuous bijection $\phi: X \rightarrow Y$ between path-connected spaces such that the induced homomorphism $\phi_{*}: \pi_1(X, x) \rightarrow \pi_1(Y, \phi(x))$ on fundamental groups is not a bijection.
Attempt: I was thinking of taking a map $$ S^1 \times \mathbb{R} \rightarrow S^1 \times S^1 $$ but I'm not sure how to get a bijection, such that the induced homomorphism is not a bijection. Any help is appreciated.
Let $X=[0,2\pi)$ and $Y=S^1$ and let $p:\mathbb{R}\to S^1$ be the quotient map. Then, $p|_X$ is continuous and bijective, but $X$ is contractible while $Y$ has a nontrivial fundamental group.