Yamagishi Fuuka
Not much is specifically said about Mitsuru and Fuuka's relationship. It is, however, thanks to Fuuka that Mitsuru can return to the field of battle. Fuuka is a much more pwerful radar and seer of shadows than Mitsuru is, Akihiko is very quickly to put this together. I believe this would cause some tension for their relationship if they had met under normal circumstances. Mitsuru served as the support before Fuuka arrived but had been doubtinf her abilities for a few months. Akihiko prompts Mitsuru to let Fuuka have her job as support because he knows that Fuuka is a much better fit. Mitsuru agrees immediately as they are being attacked by the Empress and Emperor Arcana Shadow. Mitsuru does what is best for the team, but if it hurt her or not would remain unseen.
Fuuka likes being able to be part of the team and she is grateful to have a spot on the team. She makes no decisions unless promoted by protagonist or Mitsuru. In fact, one of the older students (Akihiko or Mitsuru) needs to be available for Fuuka to even agree to take everyone into Tartarus. I think that this is the safest path Mitsuru could have them take after the rushed dedcision to include Fuuka. This still gives Mitsuru come control over what is going on, but isn't forceful mean. In fact, when asking Fuuka to join Mitsuru is upright and honest with her, stating that Fuuka has a great gift that can help people. There's a definite understanding between the two. I think that Fuuka admires Mitsuru. I also believe that in part, it would be hard for Mitsuru to have someone else "replace" her. I think she agrees because Fuuka is far more effective but I still think it would be a hard thing to come to terms with. Most likely the transition was easy because it was Akihiko's suggestion - he already knew her fears and reservations about being useless.
In fact, I think that Mitsuru probably had some reservations about allowing Fuuka to take over for her. It became less of a replacement issue than a "best for the team" idea. I think that when placed in that position, Mitsuru is safe and content to make the decision to add to the team. I think while she initally felt "replaced" or frustrated by her lack of skills, she ultimately felt more confident to assist her comrades. She does mention to Akihiko that she wished she could do more. So, I think that through Fuuka joining, Mitsuru was able to help the team much more effectively. I think that it helped to give Mitsuru focus because she was much more adept at battle than sensing shadows. I think it helped Mitsuru play to her strengths.
When Yukari asks Fuuka to find out information about an explosion that happened ten years ago, she does. This leads to Yukari finally confronting Mitsuru about Shadows, Tartarus, and their school. It leads to finding out that there was one survivor of the explosion, Mitsuru's father. It also leads the group to Yakushima to collect information. Afterwards, Mitsuru speaks to Fuuka privately and asks her to go through the Kirijo database, she compliments Fuuka's abilities. When Fuuka shows concern, Mitsuru says that she will give her her own id and pass to get into the system so that Fuuka will be free of responsibility. Fuuka agrees to help out Mitsuru because it's the least she can do to help. Fuuka feels grateful to be a valued member and because she has purpose isn't mad about the withheld information.