Is it permissible to communicate with a former foreign girlfriend via Skype to invite her to Islam after repentance and severing the relationship?

Your regret and repentance for the forbidden relationship are a great blessing that warrants gratitude and increased obedience. Satan detests a servant's repentance and strives to mislead him, so remain steadfast in your repentance and draw closer to Allah. Do not respond to the girl's messages, even under the pretext of da'wah (inviting to Islam), as that is an entry point for Satan to drag you back into the forbidden relationship. Allah Almighty says: (O you who have believed, do not follow the footsteps of Satan. And whoever follows the footsteps of Satan - indeed, he enjoins immorality and wrongdoing). If she is in need of da'wah, have one of your sisters send her addresses of Islamic or da'wah websites specifically for women, and you should ignore her messages. In fact, it is better to cancel your previous email and create a new account.