Should GIT_AUTHOR_{NAME,EMAIL} set the tagger name/email?

Maybe this is documented in some place I didn't spot, but I expected that when I set GIT_AUTHOR_{NAME,EMAIL} it would affect the operation of git-tag, but it doesn't seem to. When I create tags it seems to completely ignore those variables.

Should it be doing that? Here's a test script demonstrating the issue:

    #!/bin/sh -e
    # Set defaults
    git config --global user.name "Ævar Arnfjörð Bjarmason"
    git config --global user.email "avarab@gmail.com"

    rm -rf /tmp/test-git
    git init /tmp/test-git
    cd /tmp/test-git

    make_commit() {
        file=$1
        content=$2
        echo $content >$file
        git add $file
        git commit -m"$file: $content" $file
        git --no-pager log -1 HEAD | grep ^Author
    }

    make_commit README "testing content"
    git config user.name "Test User"
    git config user.email "test@example.com"
    make_commit README "testing content again"
    git tag -a -m"annotated tag" tag-name-1
    git --no-pager show tag-name-1 | grep ^Author

    GIT_AUTHOR_NAME="Tag Test User"
GIT_AUTHOR_EMAIL="tagtest@example.com" git tag -a -m"another annotated
tag" tag-name-2
    git --no-pager show tag-name-2 | grep ^Author

Which outputs:

    $ sh /tmp/test-tag.sh
    Initialized empty Git repository in /tmp/test-git/.git/
    [master (root-commit) 9816756] README: testing content
     1 file changed, 1 insertion(+)
     create mode 100644 README
    Author: Ævar Arnfjörð Bjarmason <avarab@gmail.com>
    [master 304b71e] README: testing content again
     1 file changed, 1 insertion(+), 1 deletion(-)
    Author: Test User <test@example.com>
    Author: Test User <test@example.com>
    Author: Test User <test@example.com>

I'd expect references to "Tag Test User <tagtest@example.com>" for the second tag I created.