Q: Is it permissible to put a photo on the internet, such as a profile picture, when a girl has hijab, complete from head to toe?
A: While such a photo is permissible because it does not display the face, to post it on facebook or any social network will not be permissible because it is propelling the female image into public limelight which itself is contrary to Islamic teaching.
For example, a woman may be completely covered, but that does not mean she is allowed to go into public places for no valid reason. The hijaab is a code of dress to be adopted when leaving home for a reason; it is not a license for the woman to go out into the public without a valid reason. The social networks are more of a danger than anything else. You might have a male attracted to this female who has posted her full hijaabi photo, and that will lead to communication and contact, which is haraam in spite of the purdah.
And Allah knows best