[FEATURE] Let new sessions join a session group at creation (inherit parent's group / assignable) instead of always landing in "No group"

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

I use the Claude Code Desktop app and organize my sidebar sessions into named session groups, one per work item (e.g. "PRC #23858", "PRC 22051", "BYD 14356"). I often create new sessions programmatically from inside a running session: an agent spawns a dedicated session (background task chip / spawn task / Dispatch) to carry out the development of a specific work item.

Every newly created session, and especially every programmatically spawned one, lands in "No group". I then have to drag it into the correct group by hand, every single time. When several work items run in parallel this is repetitive friction and the sidebar stays messy until I tidy it manually. There is no way for a spawned session to know which group it belongs to, even though it was created from a session that is already in that group.

Proposed Solution

When a new session is created from an existing one (spawn / background task / dispatch), it should by default inherit the session group of the parent session that created it.

More generally, provide a way to set the target group at creation time. Any of these would solve it:

• The spawn/task mechanism accepts a group (name or id) so the created session goes straight into it.
• A CLI flag when starting a background session, e.g. --group "PRC #23858".
• A default-group setting, or a rule that routes new sessions to a chosen group instead of "No group".
• Bring the custom named session groups that already exist in the VS Code extension (create/rename/delete, v2.1.229+) to the Desktop app, with assignment support.

Alternative Solutions

• Current workaround: manually drag each new session from "No group" into the right group after it appears. Works, but repetitive.
• Naming convention (prefixing session titles with the work item, e.g. "PRC-23858 - ...") to at least identify related sessions; they still do not group automatically.
• Desktop's automatic grouping by project does not help, because my groups are per work item, not per project (many work items share the same repo/project).

Priority

Medium - Would be very helpful

Feature Category

API and model interactions

Use Case Example

1. I'm working in a session that lives in the group "PRC #23858".
2. From that session, an agent spawns a dedicated development session (a background task chip) to implement the work item.
3. I click the chip and a new session opens.
4. Today: the new session appears under "No group", and I drag it into "PRC #23858".
5. Desired: the new session opens already inside "PRC #23858" (inherited from the parent session), or I could have specified the group when it was created.

Additional Context

• Platform: Claude Code Desktop app on Windows.
• Custom named session groups already exist in the VS Code extension (changelog v2.1.229: create/rename/delete), but assignment is manual there too, and the Desktop app does not have custom named groups at all.