Qustions concerning child custody are rarely yes or no answers. The ultimate answer in SC always lies in the "best interest of the child" standard that the Courts use to make determinations. In this question, if we presume that the mother has been adjudicated "unfit" by a court and that the grandmother has custody, the next thing we want to do is look at the father's situation. If he is a registered sex offender, what exactly does that entail? Has he been convicted for harming children, sexual assault against an adult (male or female)? Public urination or indecent exposure? It makes a difference.
This father may not get custody, but depending on his prior offense, he may be allowed some type of supervised visitation - assuming that his conviction did not involve his own child and he has been banned from contact altogether.
**This opinion is provided for information purposes only and should not be substituted for legal advice. Please seek the counsel of a licensed attorney in the appropriate jurisdiction.