Traditional IRA Recharacterization and Roth IRA Conversion at next year before extension

I have a complex IRA contribution issue:

 

on 12/12/2022: made Roth IRA contribution of $5000 for Year 2022

on 02/27/2023: made Roth IRA contribution of $1000 for Year 2022

on 04/15/2023: filed federal tax extension

on 10/01/2023: re-characterized $6000 + gain to Traditional IRA, due to income limit

on 10/16/2023: filed 2023 tax extension with recharacterization reported.

 

Then I realized I have until 11/16/2023 to file my extension because I am living in California county which was declared as national disaster area.  So I am planning to do backdoor conversion of $6000 from traditional IRA to Roth IRA.  My question is, do I need to amend 2022 return to report this conversion? If so, how? And if not, how do I report both re-characterization and the backdoor conversion in 2023 tax?