you could set up an issue security scheme (https://confluence.atlassian.com/display/JIRA/Configuring+Issue-level+Security).
Use a default security level with group A( the group to see all issues) and assignee.
You add a security level to issues not to groups. Within a security level is defined who can see an issue, so in your case you would add to this security level the Group A and the issue role assignee. In order to see an issue with this security level you must be in Group A or you must be the assignee. Please check the documentation provided.
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