A necessary and sufficient condition $h$ to lift is that $h_*(p_*(\pi_1(\tilde M)))\subseteq p_*(\pi_1(\tilde M))$. This follows from the usual conditions for a map (in this case $h\circ p:\tilde M\to M$) to lift along a covering, as given in Hatcher's book, for example. (Notice that this condition does not say that $h$ sends a $p_*(\pi_1(\tilde M))$ to a conjugate, but into itself)

A necessary and sufficient condition $h$ to lift is that $h_*(p_*(\pi_1(\tilde M)))\subseteq p_*(\pi_1(\tilde M))$. This follows from the usual conditions for a map (in this case $h\circ p:\tilde M\to M$) to lift along a covering, as given in Hatcher's book, for example. (Notice that this condition does not say that $h$ sends $p_*(\pi_1(\tilde M))$ to a conjugate, but into itself)

A necessary and sufficient condition $h$ to lift is that $f_*(p_*(\pi_1(\tilde M)))\subseteq p_*(\pi_1(\tilde M))$. This follows from the usual conditions for a map (in this case $h\circ p:\tilde M\to M$) to lift along a covering, as given in Hatcher's book, for example. (Notice that this condition does not say that $f$ sends a $p_*(\pi_1(\tilde M))$ to a conjugate, but into itself)

A necessary and sufficient condition $h$ to lift is that $h_*(p_*(\pi_1(\tilde M)))\subseteq p_*(\pi_1(\tilde M))$. This follows from the usual conditions for a map (in this case $h\circ p:\tilde M\to M$) to lift along a covering, as given in Hatcher's book, for example. (Notice that this condition does not say that $h$ sends a $p_*(\pi_1(\tilde M))$ to a conjugate, but into itself)