[FEATURE] Sidebar groups: give sessions in a group shared context (group instructions + group awareness)

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

On claude.ai/code I organize sessions into sidebar groups, one group per ongoing piece of work. When I press + next to a group, the new session shows the chip "Added to <group>", so it is clearly part of that group.

But the new session starts completely blank. It gets none of the context from the other sessions in the same group: no history, no notes, no instructions. The session can't even tell which group it belongs to, because the group isn't exposed to it. Every time I open a new session in a group, I have to explain the whole project again, even though the group already says "this belongs with that work".

Related but different: #91660 (set a group from automation) and #98439 (file new sessions into a group by folder). Both are about placing sessions in groups. This request is about what a session gets from its group.

Proposed Solution

Let a sidebar group carry shared context that every session started in that group receives automatically:

1. Group instructions: a short text field on the group (e.g. "This group is project X; read HANDOFF.md first; never deploy without approval"), injected at session start the same way CLAUDE.md is.
2. Group awareness: expose the group name to the session (e.g. in get_session / an env var), so hooks and Claude can load the right project context.
3. (Optional) Group memory: a small shared notes file that sessions in the group can read and append to, so a decision made in one session is known to the next.

Alternative Solutions

• Typing a "catch up from X" message at the start of every new session. It works, but it's manual and easy to forget, which is the problem groups should solve.
• Claude Code Projects share instructions and memory, but they are a different workflow (a coordinator conversation plus threads). The lightweight sidebar groups people already use have nothing equivalent.
• A folder-based CLAUDE.md doesn't help when every session is spawned in the same directory (e.g. one claude remote-control server).

Priority

High - Significant impact on productivity

Feature Category

Configuration and settings

Use Case Example

1. I have a group "Agents fix" holding 3 sessions about one infrastructure project.
2. I press + on that group and the new session shows "Added to Agents fix".
3. Today: the session knows nothing, and I re-explain the project, its rules and where the handoff file lives.
4. With this feature: the session starts with the group's instructions ("read the project hub and handoff first; these rules apply"), knows it's in "Agents fix", and continues the work immediately.

Additional Context

• Surface: Claude Code on the web (claude.ai/code), sessions served via Remote Control.
• Groups don't persist across devices today either (#65177), which suggests they're client-side only. Storing them server-side would be a prerequisite.