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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)