What is the ruling on an agreement between female friends to perform daily remembrance of Allah (dhikr) without restricting the number [of repetitions], with the intention of competing in good deeds and encouraging dhikr, while renewing the intention and not continuing for more than three weeks?

Cooperation in remembrance (dhikr) and mutual exhortation concerning it is not considered an innovation (bid'ah); rather, it falls under cooperation in righteousness and piety. It is preferable that your actions be discreet to avoid ostentation (riya'), unless a compelling interest necessitates making them public.