Spoiler:
I don't remember about the father and brother, but the mother pretty much knows. She's just happy to her daughter back in whatever form. FL later reveals she's not the og Irene, but they accept her anyway.
Also, Irene's soul swapped places with FL's and she's now living happily as a guide in modern Korea. Since FL and ML went back in time in addition to isekai-ing, ML didn't get tricked into crossing worlds and Irene (as FL) is paired with her own esper. Near the end, FL has this "dream" where she sees some of Irene's current life and the two talk for a bit.
I did think mother seemed to suspect something but definitely not the rest. Thanks for all the spoilers!
I kinda feel bad for the og Irene, her family doesn't even know that she is no longer with them.