Not informing Arbeitsagentur on time for starting new job

Hello, I’ve laid off in July 2023 with 3 months notice which ended at the end of October 2023. I’ve registered as unemployed with arbeitsagentur within 3 days when my contract was terminated. However, I didn’t apply for unemployment money. In December 2023, I started a new job and I thought my new employer would inform the arbeitsagentur. As I was not getting paid by arbeitsagentur at all, I only informed them about finding a new job in February 2024 when I was laid off from the new job. In February 2024, I applied for both unemployment money that i should have received for 1 month for November 2023 and from March 2024 on. But arbeitsagentur paid me for December, January and February too so I received extra 3 months money which then they asked me to send them back and I did immediately. However, now they started an investigation against me and asked me to fill a form to explain the situation. Is there anyone who had a similar issue before? Should I admit in this form that I made an infringement?

thank you!