[rush] Write last-install.flag after update-autoinstaller completes

[Copilot]

Summary

Fixes #5464

rush update-autoinstaller performs installation but does not write the last-install.flag file, causing subsequent rush install-autoinstaller to redundantly reinstall.

Details

Added _createFlagsAsync() method to Autoinstaller.ts that writes both:

• .rush/temp/last-install.flag - tracks install state (node version, package manager, package.json contents)
• node_modules/rush-autoinstaller.flag - marker file for node_modules validity

Called at the end of updateAsync() after successful installation, matching the behavior of prepareAsync().

How it was tested

• Built rush-lib and verified compilation
• Ran full test suite (528 tests pass)

Original prompt

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This section details on the original issue you should resolve

<issue_title>[rush] Can update-autoinstallers write common/autoinstallers/rush-plugins/.rush/temp/last-install.flag?</issue_title>
<issue_description>## Summary

After executing rush update-autoinstaller, the .rush/temp/last-install.flag file was not written, causing subsequent related commands to perform the installation again.

Repro steps

• Change any dependencies version in autoinstallers/*/package.json
• Run rush update-autoinstaller --name xxx.
• Run rush install-autoinstaller --name xxx. Expect to skip installation.

Standard questions

Please answer these questions to help us investigate your issue more quickly:

Question	Answer
@microsoft/rush globally installed version?	5.155.1
rushVersion from rush.json?	5.155.1
useWorkspaces from rush.json?	Yes
Operating system?	Mac
Would you consider contributing a PR?	Yes
Node.js version (node -v)?	22.20.0
</issue_description>

<agent_instructions>Running rush update-autoinstaller performs installation, so the last-install flag should be written when it completes, same as if the autoinstaller is automatically installed, or if the user runs rush install-autoinstaller.</agent_instructions>

Comments on the Issue (you are @copilot in this section)

@iclanton Looks like a simple oversight. @LPegasus, that seems like the right place.

• Fixes [rush] Can update-autoinstallers write common/autoinstallers/rush-plugins/.rush/temp/last-install.flag? #5464

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[iclanton]

@copilot the description mentions the wrong issue. It should mention #5464

[Copilot]

@copilot the description mentions the wrong issue. It should mention #5464

Updated the PR description to reference issue #5464.