How to setup a trigger for updated records

Prod database "Mother" Copy of database "Father"

Here's my scenario I need help with: I have a database that has a table called User_Group table on our Production db "mother". The dev team wants me to add a trigger to add those changes into the same table on another database "Father"(on another server). The the remote database is "Father" and when the so when records get changed on "Mother" they are copied into there?

Does that make sense? Any ideas?

I suggested replication but they do not want that.