"Single" and "Head of Household" are two different filing statuses. Which did you use on the federal return? You can look near the top of your Form 1040 to see, and you can also see the dependents you claimed near the top of that form. The information flows from the federal to the state form, so if you have HOH and the dependents on your federal form it should show up on the state form too.
PREVIEW 1040
https://ttlc.intuit.com/questions/1901539-how-do-i-preview-my-turbotax-online-return-before-filing
Am I Head of Household?
https://ttlc.intuit.com/questions/1894553-do-i-qualify-for-head-of-household
https://ttlc.intuit.com/questions/2900097-what-is-a-qualifying-person-for-head-of-household
California has stricter rules for Head of Household. On federal if you are caring and paying for a child who lives with you, but with whom you have no relation to (eg, a significant other's child from a previous marriage) you qualify as Head of Household,. In California, you would not be able to claim HOH for that child you are supporting. So what to do if the requirements for status are not the same for state and federal? It's a conundrum. Maybe that's why his dependents don't show up on his California return.
Sorry but you are mistaken about the rule for claiming a dependent on a federal return in order to file HOH. The child must be related to you ---- the child of a girlfriend or boyfriend is not a qualifying dependent for you to file as HOH. It has to be YOUR child or blood-related relative. So you are probably making a mistake on your federal return.
Am I Head of Household?
https://ttlc.intuit.com/questions/1894553-do-i-qualify-for-head-of-household
https://ttlc.intuit.com/questions/2900097-what-is-a-qualifying-person-for-head-of-household
If you qualify as Head of Household, when you enter your filing status (single or married filing separately) into MyInfo, and then enter your qualifying dependent, TurboTax will offer HOH as your filing status.
@MikeFD The child of a "significant other", GF or BF's child who lives with you cannot ever be a qualifying dependent that enables you to file as Head of Household or to get the child related credits like earned income credit, child tax credit or stimulus payments. All you "get" for claiming a child who is unrelated to you who lived with you for the whole year is the $500 credit for other dependents.