Is it permissible for him to enable his sister to see her suitor outside the home without his father’s knowledge?
It is prescribed for the suitor to see his potential fiancé and to be allowed to do that; it is also prescribed for her to be allowed to see him, so that if there is agreement to go ahead with the marriage it is done on the basis of clarity and insight.
Abu Dawood (2082) narrated that Jaabir ibn ‘Abdillah said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “When one of you wants to propose marriage to a woman, if he can see that which will encourage him to go ahead and marry her, let him do so.” I proposed marriage to a girl and I used to hide and watch her until I saw that which prompted me to go ahead and marry her, and I married her.
Classed as hasan by al-Albaani in Saheeh Abi Dawood
He may see that of her that which will prompt him to go ahead and marry her, such as that which ordinarily appears such as the face, hands, neck and feet.
It says in ar-Rawd al-Murbi‘ (p. 332):
It is permissible for the one who wants to propose to a woman and thinks it most likely that she will accept, to look at what ordinarily appears, such as the face, necks, hands and feet. End quote.
The Scholars of the Standing Committee said:
It is permissible for the one who wants to marry a woman to look, when proposing to her, at her face without any feelings of pleasure or desire, and without being alone with her, according to scholarly consensus. That is prescribed because of necessity and because it is more likely to make their marriage stable. And that is sufficient because the face is the focal point of beauty and by means of this the need is met. And some scholars regarded it as permissible to look at the hands too, and whatever ordinarily appears of the woman that may encourage him to go ahead and marry her. And it is permissible for the suitor to watch her when she is walking in the street so that he may see that which may encourage him to go ahead and marry her.
End quote from Fataawa al-Lajnah ad-Daa’imah, 18/75
See also the answer to question no. 2572
If the matter is as you describe, there is nothing wrong with you enabling your sister to see this suitor and enabling him to see her, even if that is outside the home, on condition that you or a mahram of hers is present during the meeting. What matters is that he should not be alone with her and there should be no cause of fitnah, as the scholars have explained previously.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
He may look at the face, head, hands and feet, on condition that he is not alone with her; rather one of her mahrams should be present. It is also essential that there be no risk of fitnah. He should look only as much as is necessary, but if he is not able to see her the first time, there is nothing wrong with him looking at her another time, but that is subject to the condition we have mentioned, which is that he should not be alone with her, because being alone with a woman is haraam. Similarly, she should not beautify or adorn her face when he looks at her, because that will have the opposite effect. If he looks at her when she has beautified herself he will think that she is more beautiful than she really is, then when he enters upon her (after marriage) and sees her as she really is, he may be put off and turn away from her.
End quote from Fataawa Su’aal min Haaj (p. 39)
But you have to make sure your father does not find out about that, so that he will not get angry or refuse to let her marry him, when the marriage is agreed upon.
If it is possible to make him agree and convince him of that, then that is better and is preferable.
We should also point out that if the passing glance you mentioned was sufficient for your sister and was also sufficient for the suitor to form an opinion and lead to an agreement, then this is what is required and there is no need to look directly.
And Allah knows best.