Is it permissible to count the amount paid to help one's stepmother perform the Hajj as being from Zakat funds?

It is permissible to give Zakat to a poor stepmother, as she is among those eligible to receive it, provided that her expenses are not covered by a husband or anyone else, and that giving her Zakat is not intended as a way to avoid a customary obligation of financial assistance. However, if she is not eligible for Zakat, then what is given to her cannot be counted as Zakat; rather, it is considered an act of charity for which the Muslim will be rewarded according to his intention.