Hide the assignee and use your own custom user-picker. You'll probably find you don't want to do it though, the assignee field is heavily used by all sorts of stuff in Jira, so you'll be sacrificing a lot of reporting and feedback options.
I don't think that's really what you want to do though. Could you explain why you think you want to do this? There's probably a much better way to do what you need.
I need in dependency of the current user different content in the field assignee.
example:
current user = tester1 ==> field assignee contains all members of group tester
current user = developer1 ==> field assignee contains all members of group developer
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I'm not sure I understand that. Do you mean you want to limit the list of possible assignees to a group, depending on the current user?
If that's the case, you can't do it with any built-in field, and removing the assignee field is a red-herring. This is quite a complex requirement, and it's going to need some coding.
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