The instructions for the form are usually a good place to start.
If you file Form 1040, 1040-SR, 1040-NR, 1040-SS, or 1041
for 2019, remember to attach Schedule H to it. Mail your return,
by April 15, 2020, to the address shown in your tax return
Exceptions. If you get an extension of time to file your return,
file your return with Schedule H by the extended due date. If
you’re a fiscal year filer, file your return and Schedule H by the
due date of your fiscal year return, including extensions.
Schedule H can also be filed stand-alone (as mentioned in the instructions). No stand-alone extension is mentioned.
Since most of the payment due with a stand-alone Schedule H is trust fund tax money, it doesn’t seem like a payment extension is really warranted, but to be safe, why not just file it with a Form 1040 (even if not required)? Then you can extend and take full advantage of other changes for this filing season.