"I do not care about František."
Translation:O Františka se nestarám.
The notes at the start of the unit say "se" wants to be after the second unit of meaning, so it kind of depends on how you word your sentence (since word order can be all loosey goosey in Czech). I saw a couple examples where the verb came first and "se" came immediately after and other examples where the object came first and then "se" came afterwards. So whether or not it precedes the verb depends on where the verb comes in the sentence, I think: if the verb is the first unit of meaning in the sentence, "se" comes after it, but if the verb comes later in the sentence "se" will come before it.
So... this verb means both "to take care of" and "to care about"? Those mean two different things in English. One might, for example, "take care of the pigs" (providing physical care to them) without actually "caring about" the pigs (having some concern for their welfare)..