(Lifting criterion) Suppose given a covering space $p:(\tilde{X},\tilde{x}_0)\to (X,x_0)$ and a map $f:(Y,y_0)\to (X,x_0)$ with $Y$ path-connected and locally path connected. Then a lift $\tilde{f}:(Y,y_0)\to (\tilde{X},\tilde{x}_0)$ of $f$ exists iff $f_* (\pi_1 (Y,y_0))\subset p_* (\pi_1 (\tilde{X},\tilde{x}_0))$.
My question is that: Does this theorem still hold if $f_* (\pi_1 (Y,y_0))\subset p_* (\pi_1 (\tilde{X},\tilde{x}_0))$ is replaced by $f_* (\pi_1 (Y,y_0))\cong \leqslant p_* (\pi_1 (\tilde{X},\tilde{x}_0))$?
The answer is no. Take as an example $Y$ a torus with $\pi_1(Y)=\langle a,b\mid aba^{-1}b^{-1}\rangle$, and let $\tilde{Y}$ be the cover associated to the subgroup $\langle b\rangle$, ie an infinite cyclinder. Then $\pi_1(\tilde{Y})$ is isomorphic to $\mathbb{Z}$ and its image under $p_\ast$ is $\langle b\rangle<\langle a,b\mid aba^{-1}b^{-1}\rangle$.
Now take as your space $X$ the circle which has $\pi_1(X)=\mathbb{Z}$, and let $f$ map this circle to a loop representing the element $a$ in $\pi_1(Y)$. Then $f_\ast(\pi_1(X))$ is $\langle a\rangle$ which is isomorphic to $p_\ast(\pi_1(\tilde{Y}))=\langle b \rangle$, but there is no lift.