Should he wake his family for Fajr prayer even though they are exhausted and tired?
Question

                Is it permissible for me to wake my wife for Fajr prayer knowing that she breastfeeds her small daughter during the night and sometimes she is very exhausted?.
Praise be to Allah.Prayer is very important, 
    and we are commanded to offer prayers at the prescribed times; we are urged 
    to do that and we are warned against neglecting it. This is well known. For 
    example, Allaah says (interpretation of the meaning): 
“Verily, As‑Salaah (the 
    prayer) is enjoined on the believers at fixed hours”
[al-Nisa’ 4:103] 
“Guard strictly (five 
    obligatory) As‑Salawaat (the prayers) especially the middle Salaah (i.e. the 
    best prayer ‑ ‘Asr). And stand before Allaah with obedience”
[al-Baqarah 2:238]
And the Prophet  (peace 
    and blessings of Allaah be upon him) said, when he was asked which deed is 
    most beloved to Allaah: “Prayer offered on time.” Narrated by al-Bukhaari 
    (527) and Muslim (85). 
So you should beware of 
    missing prayer or delaying it until the time for it is over, because Allaah 
    says (interpretation of the meaning): 
“Then, there has 
    succeeded them a posterity who have given up As‑Salaah (the prayers) [i.e. 
    made their Salaah (prayers) to be lost, either by not offering them or by 
    not offering them perfectly or by not offering them in their proper fixed 
    times] and have followed lusts. So they will be thrown in Hell”
[Maryam 19:59]
It is not permissible to 
    delay prayers beyond their appointed time unless that is for a reason that 
    makes it permissible to join prayers at the time of the earlier or later 
    prayer. It is well known that Fajr prayer cannot be joined with any prayer 
    before or after it, so it must be offered on time, no matter how sick or 
    tired a person is, so long as he is conscious and of sound mind. He should 
    pray standing, but if he cannot then he may offer it sitting, and if he 
    cannot do that then he may pray lying down.  
The sleeper should take 
    precautions to help him wake up, such as not staying up late, setting an 
    alarm clock and asking others to help him wake up. 
The husband is enjoined to 
    take care of his family and save them from the Fire, as Allaah says 
    (interpretation of the meaning):  
“O you who believe! Ward 
    off yourselves and your families against a Fire (Hell) whose fuel is men and 
    stones, over which are (appointed) angels stern (and) severe, who disobey 
    not, (from executing) the Commands they receive from Allaah, but do that 
    which they are commanded”
[al-Tahreem 66:6]
Based on this, you should 
    be keen to wake up your wife and encourage her and help her to pray; you can 
    delay waking her up until it is nearly the end of the time for prayer. We 
    ask Allaah to guide and help us and you. 
And Allaah knows best.