Firstly, we’d like to make it clear that wiping over the head is one of the pillars of Wudu’. Scholars say that one may wipe over even few hairs. Wudu’ without wiping over the head is invalid. So, you need to rub even some hairs under your hijab or headscarf with water.
Elaborating on this, the eminent Muslim scholar, Dr. `Abdul-Fattah `Ashoor, Professor of the Exegesis of the Qur’an at Al-Azhar Univ., states: “wiping over the head is one of the obligatory acts of Wudu’. Some scholars say that one may wipe over some hairs.
Allah says in the Qur’an, “O you who believe, when you rise for prayer, wash your faces and your hands up to the elbows and lightly rub your heads and (wash) your feet up to the ankles.” (Al-Mai’dah: 6)
accordingly, you, dear sister, have to wipe over even some hairs from underneath the headscarf.”
Based on the above Fatwa, it becomes crystal clear that rubbing some hairs with water is a must in order for Wudu’ to be valid. Also, one cannot rub the headscarf, for the Sunnah permitted Muslims to wipe over socks, not headscarves.

It should also be clear that wiping over socks is a general dispensation that is not peculiar to the season of winter as some people think. It is permissible any time throughout the year for travelers as well as for residents. But there is a duration of wiping to be stuck to by residents and travelers. The residents are allowed to wipe over their socks for one day and night, while the travelers are allowed to wipe over them for three days and nights.