Suppose that $\pi_{1}\left(Y, y_{0}\right)$ is finitely presented. How can we construct a path connected finite cellular space $\left(X, x_{0}\right)$ and a continuous map $\psi:\left(X, x_{0}\right) \rightarrow\left(Y, y_{0}\right)$ such that the induced homomorphism $$ \pi_{1}(\psi): \pi_{1}\left(X, x_{0}\right) \rightarrow \pi_{1}\left(Y, y_{0}\right) $$ is an isomorphism.
I think we have to use this theorem, but the construction part is not clear to me: If $f:\left(X, x_{0}\right) \rightarrow\left(Y, y_{0}\right)$ is a homotopy equivalence of spaces with base points, then $f_{*}: \pi_{1}\left(X, x_{0}\right) \rightarrow \pi_{1}\left(Y, y_{0}\right)$ is an isomorphism.