I run a secret Facebook group. If I need to add a new member, I'll add an email address in the Invite Members field in the right pane:
Sometimes it works as expected.
In 90% of the cases in which I send an invitation, the intended recipients tell me that they have never received a notification or email from Facebook about the invitation, and therefore remain excluded from the group.
Then my fallback approach (the only sure way I invite someone into the secret group) is to first connect with them as a "friend." then Insert your e-mail address in this field Invite members. That always works. (But I do not want to connect as friends.)
This problem has existed since last year.
I wonder if that worked a few times due to certain Facebook profile / privacy / notification settings or preferences of the people I invited.
If I could narrow down to understand which settings allow someone to seamlessly receive invitations, I could direct potential members to adjust their settings first.