You theoretically can... but the fire would still be hot.
From Mrs Weasley giving Mr Diggory some toast (quoted in another answer but repeated here for ease of reading), we can see the head in the fireplace has a physical presence.
"Never mind, Amos,” said Mrs Weasley. “Sure you won’t have a bit of toast or anything before you go?”
“Oh go on, then,” said Mr Diggory.
Mrs Weasley took a piece of buttered toast from a stack on the kitchen table, put it into the fire tongs, and transferred it into Mr Diggory’s mouth.
“Fanks,” he said in a muffled voice, and then, with a small pop, vanished.
– Harry Potter and the Goblet of Fire, Chapter 11 (Aboard the Hogwarts Express)
However, the fire around the head in the fireplace was still hot. Crookshanks tried to get close to Sirius but couldn’t because of the heat from the fire.
“Sirius!’ he said.
Harry whipped round. Sirius’s untidy dark head was sitting in the fire again.
‘Hi,’ he said, grinning.
‘Hi,’ chorused Harry, Ron and Hermione, all three kneeling down on the hearthrug. Crookshanks purred loudly and approached the fire, trying, despite the heat, to put his face close to Sirius’s.” - Harry Potter and the Order of the Phoenix, Chapter 17 (Educational Decree Number Twenty-four)
So, the head has a physical presence, but people can’t just reach out and touch it - they’ll get burned.
Umbridge likely transported just her hand using Floo Powder.
When Umbridge’s hand came through the fireplace, it seems most logical that she did that by reaching into a Floo Powder fire and transporting just her hand over the Floo Network, similar to how people can stay where they are and transport just their heads.
‘Number twelve, Grimmauld Place!’ Harry said loudly and clearly.
It was one of the most curious sensations he had ever experienced. He had travelled by Floo powder before, of course, but then it had been his entire body that had spun around and around in the flames through the network of wizarding fireplaces that stretched over the country. This time, his knees remained firm upon the cold floor of Umbridge's office, and only his head hurtled through the emerald fire …
- Harry Potter and the Order of the Phoenix, Chapter 29 (Careers Advice)
From what we see of how Floo transport works, it seems likely that Umbridge stuck her hand in the fireplace and transported it alone.
“Hermione gave a horrified gasp and leapt to her feet, still staring at the fire.
A hand had appeared amongst the flames, groping as though to catch hold of something; a stubby, short-fingered hand covered in ugly old-fashioned rings.
The three of them ran for it. At the door of the boys’ dormitory Harry looked back. Umbridge’s hand was still making snatching movements amongst the flames, as though she knew exactly where Sirius’s hair had been moments before and was determined to seize it.”
- Harry Potter and the Order of the Phoenix, Chapter 17 (Educational Decree Number Twenty-four)
She probably could have pulled Sirius through the Floo Network.
It seems likely that, if she had been able to grab ahold of Sirius (her hand being in a Floo fire, and not a hot one), she’d be able to pull the rest of him through the Floo Network by his head.