Theory:

She had a scarlet cap on her head,
And that was left the same;
But all the rest of her clothes were burned
Black as a coal in the flame.

And every country schoolboy
Has seen her in the wood,
Where she lives in the trees till this very day,
Boring and boring for food.
Explanation:
 
When the woman came out through the chimney as a bird, she became completely transformed and unrecognisable. Everything was changed, except for the scarlet cap she had on her head. Her clothes were burned as she came through the burning chimney. They had turned as black as coal because of the soot from the chimney.

Dryocopus_martius gertrude's bird.jpg
The woman was turned into a woodpecker with black body and scarlet head

It is interesting to note how the speaker has associated the appearance of a woodpecker with the attire of the woman. It is natural for a woodpecker to have a scarlet head and black body. However, the speaker claims that the bird was turned black because of the flame and soot, and the scarlet head represents the colour of the cap the woman was wearing.

As we come to the final stanza of the poem, the speaker prepares to conclude the ballad. She says that the woman still lives in the form of a bird. Moreover, every schoolboy has seen her in the trees as they crossed the forest. And, as Saint Peter had cursed, she keeps "boring and boring for food".
 
forest-3394066_1920.jpg
She still lives in the trees boring and boring for food

In the end, the woman who couldn't understand or empathise with another person's hunger and misfortune had ended up learning it through a hard lesson.
 
Meanings of difficult words:
 
S.No
Words
Meanings
1
Scarlet(Of colour) bright red
2
SootA black powder produced when coal, wood, etc., is burned
Reference:
National Council of Educational Research and Training (2006). Beehive. A Legend of the Northland - Phoebe Cary (pp. 65 - 67). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.