Genes that control hair or feather color in some animals are expressed differently in the winter than the summer. how might such a difference be beneficial to the ptarmigan shown here

I know that the environment plays a role in the changing of hair or feathers in some animals so here: 

Winter- color changes to more resemble winter surroundings. You'll see more white colors to blend in. 

Summer- the colors more resemble the summer colors; you'll see more browns and rocky looking colors to hide/blend in. 

Spring- you'll see very vibrant colors like that of a flower.

Those are some examples, but I hope this helps. 

This movement is also referred to as  glacial plucking