diff --git a/cronic b/cronic
index 1bb3849..1644dcf 100755
--- a/cronic
+++ b/cronic
@@ -68,6 +68,9 @@ cronic_help() {
   echo "stderr, from which lines that start with PS4 are removed.  That removal"
   echo "is skipped if PS4 contains \"\$\", a backquote, a backslash, or a newline,"
   echo "because such a PS4 can expand differently on each line."
+  echo "Do not export the verbose option (\"set -v\") via SHELLOPTS: bash would echo"
+  echo "the lines of this script and of a bash COMMAND to stderr, so every run would"
+  echo "produce output."
   exit 0
 }
 
diff --git a/tests/cronic-test/cronic-test.sh b/tests/cronic-test/cronic-test.sh
index 52f89ee..6d8f9fb 100755
--- a/tests/cronic-test/cronic-test.sh
+++ b/tests/cronic-test/cronic-test.sh
@@ -341,51 +341,43 @@ check "EXIT trap writes to stdout" 0 silent trap 'echo "from the trap"' EXIT
 check "builtin with an unset variable" 0 silent \
   eval ': "$cronic_test_unset_variable"'
 # When the caller exports SHELLOPTS, a bash script runs with the caller's
-# options rather than with `cronic`'s own `set -eu`.  BASH_ENV is empty so that
-# the script runs no startup file.
+# options rather than with `cronic`'s own `set -eu`.
 # Creates "$work/cronic-with-NAME", which runs `cronic` with SHELLOPTS exported,
-# after `set OPTION` if OPTION is given.
+# after `set OPTION` if OPTION is given.  The wrapper unsets BASH_ENV, so that
+# neither `cronic` nor a bash script that it runs reads the user's startup
+# file.  The wrapper's own trace goes to /dev/null, so that with `set -x`, any
+# trace in the output comes from `cronic`.
 make_cronic_wrapper() { # NAME [OPTION]
-  {
-    echo "#!/bin/bash"
-    if [ $# -gt 1 ]; then
-      echo "set $2"
-    fi
-    echo "export SHELLOPTS"
-    echo "exec \"$CRONIC\" \"\$@\""
-  } > "$work/cronic-with-$1"
-  chmod +x "$work/cronic-with-$1"
-}
-make_cronic_wrapper shellopts
-make_cronic_wrapper nounset -u
-make_cronic_wrapper errexit -e
-make_cronic_wrapper allexport -a
-# The wrapper's own trace goes to /dev/null, and `cronic` reads no startup
-# file, so that any trace in the output comes from `cronic`.
-cat > "$work/cronic-with-xtrace" << EOF
+  cat > "$work/cronic-with-$1" << EOF
 #!/bin/bash
 unset BASH_ENV
 exec 3>&2 2> /dev/null
-set -x
+${2:+set $2}
 export SHELLOPTS
 exec "$CRONIC" "\$@" 2>&3 3>&-
 EOF
-chmod +x "$work/cronic-with-xtrace"
+  chmod +x "$work/cronic-with-$1"
+}
+make_cronic_wrapper shellopts
+make_cronic_wrapper nounset -u
+make_cronic_wrapper errexit -e
+make_cronic_wrapper allexport -a
+make_cronic_wrapper xtrace -x
 saved_cronic="$CRONIC"
 CRONIC="$work/cronic-with-shellopts"
 # shellcheck disable=SC2016 # The expansion is for the wrapped command to do.
 check "bash script with an unset variable, SHELLOPTS exported" 0 silent \
-  env BASH_ENV= bash -c ': "$cronic_test_unset_variable"'
+  bash -c ': "$cronic_test_unset_variable"'
 check "bash script with a failing command, SHELLOPTS exported" 0 silent \
-  env BASH_ENV= bash -c 'false; true'
+  bash -c 'false; true'
 CRONIC="$work/cronic-with-nounset"
 # shellcheck disable=SC2016 # The expansion is for the wrapped command to do.
 check "bash script with an unset variable, caller's nounset exported" 127 \
   "report:cronic_test_unset_variable" \
-  env BASH_ENV= bash -c ': "$cronic_test_unset_variable"'
+  bash -c ': "$cronic_test_unset_variable"'
 CRONIC="$work/cronic-with-errexit"
 check "bash script with a failing command, caller's errexit exported" 1 report \
-  env BASH_ENV= bash -c 'false; true'
+  bash -c 'false; true'
 # When the caller exports allexport, `cronic` does not export its own variables
 # to the command, but a variable that the command assigns is exported.
 CRONIC="$work/cronic-with-allexport"
@@ -397,9 +389,9 @@ check "command's variable exported, caller's allexport exported" 0 silent \
 # When the caller exports xtrace, `cronic` traces only the command, not itself.
 CRONIC="$work/cronic-with-xtrace"
 check "bash script, caller's xtrace exported" 0 silent \
-  env BASH_ENV= bash -c 'true'
+  bash -c 'true'
 check "bash script that fails, caller's xtrace exported" 1 "report:^+ false$" \
-  env BASH_ENV= bash -c 'false'
+  bash -c 'false'
 if grep -q cronic_ "$work/output"; then
   echo "FAIL: caller's xtrace exported: the report traces cronic itself:"
   cat "$work/output"