All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
First of all, we draw attention to the fact that adoption with the meaning of tracing back a child to other than his/her father is Islamically forbidden, but if the meaning is just to be kind to the child while not changing his/her family lineage, then this is Islamically permissible and it is an act by which one gets closer to Allaah and gets a reward from Him. We have already clarified this in Fatwa 82371.
Being concerned about the physical education of that girl is something good but what is more important than this is spiritual education and nurturing her on the teachings and principles of Islam especially that she is at an age when she needs guidance to good moral standards and protection from reasons of temptation and corruption.
For more benefit on some educational guidelines for the children, please refer to Fataawa 86237 and 89684. Also, for more benefit on teaching a girl the importance of Hijab and its rulings, please refer to Fatwa 101325.
Based on that, this couple (this husband and wife) should be reminded that this girl is a trust with them and safeguarding the trust is Islamically required; rather, it is an obligation. Since this is the case in regard to the general trusts, so when the trust is a girl, the matter is even more confirmed in her regard.
If they cannot nurture her by themselves, then they should bring for her a righteous woman who would nurture her or they should give her back to her family (parents). However, leaving her like this without an educator and without an instructor with the pretext of having trust in Allaah and relying on Him, is a strange matter; rather it is Tawaakul (i.e. putting one's trust in Allaah without taking any necessary means) and not Tawakkul (i.e. taking the proper means while putting one's trust in Allaah). Indeed, Tawakkul means taking the proper means while putting the matter in the Hands of Allaah.
On the other hand, if this couple spend on themselves in dispraised extravagance, then one should advise them and remind them of Allaah and of the verses that dispraise being extravagant. Allaah Says (what means): {O children of Adam, take your adornment [i.e. wear your clothing] at every Masjid, and eat and drink, but be not excessive. Indeed, He likes not those who commit excess.} [Quran 7:31]
Nonetheless, it should be noted that extravagance is a relative matter in the sense that what is considered as extravagance for some people because they are poor is not considered as extravagance for others who are wealthy.
Also, it should be noted that this girl is a non-Mahram (a marriageable woman) for the man who is nurturing her, so that man is not permitted to be in seclusion with her unless his wife or his sister or his mother had breastfed that girl five times while she was below the age of two years.
Allaah Knows best.