[BUG] Scheduled task's browser tool restricted to one domain despite "On all sites" extension permission

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

Claude in Chrome's extension "Site access" is set to "On all sites," but browser navigation
during a scheduled/background task run is restricted to a single previously-used domain
(redfin.com in my case). Attempts to navigate to any other domain — including unrelated ones
like google.com and github.com — fail with "Navigation to this domain is not allowed."

This does NOT reproduce in a normal interactive Claude Code session with the same extension
and account — those domains navigate fine there. It only shows up when the task originates
from a scheduled task trigger.

Repro:
1. Set up a scheduled task that uses Claude in Chrome to browse multiple domains.
2. Let it fire on schedule (unattended).
3. Observe that only domains previously visited in an interactive session succeed; any new
domain is rejected outright, with no permission prompt shown.

Expected: scheduled/background browser sessions should have the same site access as an
interactive session when the extension is set to "On all sites."

### What Should Happen?

I want this fixed

### Error Messages/Logs

```shell

```

### Steps to Reproduce

i want the claude to be able to have access to all ites and not be restricted

### Claude Model

None

### Is this a regression?

I don't know

### Last Working Version

_No response_

### Claude Code Version

claude --version 2>&1

### Platform

Anthropic API

### Operating System

macOS

### Terminal/Shell

Terminal.app (macOS)

### Additional Information

_No response_

No repository file or test is identified in the report. Start by reproducing the scheduled-task trigger with Claude in Chrome and compare navigation to an interactive session while Site access is set to "On all sites." Done means scheduled/background sessions can navigate to previously unused domains without the "Navigation to this domain is not allowed" error.

索引モデルが issue の本文から書いたものです。