diff --git a/Lib/tarfile.py b/Lib/tarfile.py
index a0fab46b24e249..75de0e3a1d063a 100644
--- a/Lib/tarfile.py
+++ b/Lib/tarfile.py
@@ -480,25 +480,27 @@ def _init_read_gz(self):
         # taken from gzip.GzipFile with some alterations
         if self.__read(2) != b"\037\213":
             raise ReadError("not a gzip file")
-        if self.__read(1) != b"\010":
+        (method, flag, _) = struct.unpack("<BBIxx", self.__read(8))
+        if method != 8:
             raise CompressionError("unsupported compression method")
 
-        flag = ord(self.__read(1))
-        self.__read(6)
-
+        # process FEXTRA
         if flag & 4:
-            xlen = ord(self.__read(1)) + 256 * ord(self.__read(1))
-            self.read(xlen)
+            extra_len, = struct.unpack("<H", self.__read(2))
+            self.__read(extra_len)
+        # process FNAME
         if flag & 8:
             while True:
                 s = self.__read(1)
-                if not s or s == NUL:
+                if not s or s == b'\000':
                     break
+        # process FCOMMENT
         if flag & 16:
             while True:
                 s = self.__read(1)
-                if not s or s == NUL:
+                if not s or s == b'\000':
                     break
+        # process FHCRC
         if flag & 2:
             self.__read(2)
 
diff --git a/Lib/test/archivetestdata/tgz_with_comment_extra_data_in_header.tgz b/Lib/test/archivetestdata/tgz_with_comment_extra_data_in_header.tgz
new file mode 100644
index 00000000000000..b3d5ca05c1f94d
Binary files /dev/null and b/Lib/test/archivetestdata/tgz_with_comment_extra_data_in_header.tgz differ
diff --git a/Lib/test/test_tarfile.py b/Lib/test/test_tarfile.py
index 54d329a15d4d25..eadf83fecdc914 100644
--- a/Lib/test/test_tarfile.py
+++ b/Lib/test/test_tarfile.py
@@ -45,6 +45,7 @@ def sha256sum(data):
 TEMPDIR = os.path.abspath(os_helper.TESTFN) + "-tardir"
 tarextdir = TEMPDIR + '-extract-test'
 tarname = support.findfile("testtar.tar", subdir="archivetestdata")
+tgzname_with_comment_extra_data_in_header = support.findfile("tgz_with_comment_extra_data_in_header.tgz", subdir="archivetestdata")
 gzipname = os.path.join(TEMPDIR, "testtar.tar.gz")
 bz2name = os.path.join(TEMPDIR, "testtar.tar.bz2")
 xzname = os.path.join(TEMPDIR, "testtar.tar.xz")
@@ -859,6 +860,11 @@ def test_read_through(self):
                     if not buf:
                         break
 
+    @unittest.skipIf(zlib is None, "requires zlib")
+    def test_read_with_extra_header(self):
+        with tarfile.open(tgzname_with_comment_extra_data_in_header,
+                          mode="r|*") as _:
+            pass
     def test_fileobj_regular_file(self):
         tarinfo = self.tar.next() # get "regtype" (can't use getmember)
         with self.tar.extractfile(tarinfo) as fobj:
diff --git a/Misc/NEWS.d/next/Library/2024-11-02-02-02-31.gh-issue-107398.uUtA6Q.rst b/Misc/NEWS.d/next/Library/2024-11-02-02-02-31.gh-issue-107398.uUtA6Q.rst
new file mode 100644
index 00000000000000..728784cb42c5c1
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2024-11-02-02-02-31.gh-issue-107398.uUtA6Q.rst
@@ -0,0 +1,2 @@
+Fix :mod:`tarfile` stream mode exception when process the file with extra header.
+data.