When she first meets Kouichi, she was very warm and welcoming to him. Upon attending school however she was very cold and strict to him, giving him cryptic warnings and not sharing any vital information to him. She didn't start to truely befriend him until episode 7 where she invites him to a cafe. She then playfully teases him into thinking he is the Extra, before backing off when they learn from Mochizuki of a crucial part of the calamity. This is not all one sided as he playfully banters back at her in episode 8.
It was hinted througout the later episodes that Izumi may have had a romantic interest in Kouichi. This was mostly due to the fact that she feels like she met him, at some point before the events of Class 3-3. An event she barely remembers, but Kouichi didn't remember at all. Upon her death in episode 12 she remembers accidentally hitting him with an empty Coffee Soda can. When she went to see when he was alright she tripped and fell stumbling down a hill. Kouichi helps her up and introduced himself shaking her hand. This act is what sparked her interest in him.
At the start Izumi was warm and welcoming to Kouichi, but upon entering Yomi North she acted cold towards him. Berating him for vioating rules he was unaware of violating simply for talking to Mei who was the Non-Existent Student at the time. In episode 7 she flat out stated that the Calamity was his fault and Mei's. The hate ship is not one sided though, for in episode 10 he defended Mei from her accusations. An act that caused Izumi to bite her lip in anger. In episode 12 she was willing to kill Kouichi when he tried to stop her from killing Mei. However she is killed when a lighting strike hits the building and sends glass and debris into her body.