What does Lina not tell her mother about Kostas in Between Shades of Gray?

The answer to your question can be found in Chapters 12 and 13 in the book.
In Chapter 12, the Vilkas children (Lina and Jonas) and Andrius Arvydas (their friend and another Lithuanian deportee) finally catch up with Kostas Vilkas.
Kostas is the father of Lina and Jonas. In Chapter 12, new cattle wagons begin lining up behind the trains carrying Lina, Jonas, Andrius and other Lithuanian deportees. The children go from train to train to try and see if they can find Kostas Vilkas and Andrius's father, Petras Arvydas.
They finally come across Kostas in one of the trains. When they see him, they notice that his face is ashen and one of his eyes is badly bruised. From all appearances, Kostas has been beaten. This is what Lina neglects to tell her mother, Elena.
During their interactions with their father, Lina and Jonas learn that they are all going to Siberia. Accordingly, Kostas's train is to be attached to the ones carrying Lina, Jonas, Elena, Andrius, and the other Lithuanian deportees.
Before they part ways, Kostas hands his children two jackets, some shirts, and socks. He also gives them a piece of ham and tells them to divide it among themselves. Finally, Kostas gives Lina his gold wedding band: it is for Elena, in case she needs to sell it for necessities.
As for Lina and Jonas, both are grieved that they must leave their father. Kostas, however, tells them to have courage and to tell their mother to remember the oak tree. He also maintains that they can help him find them in due time.
In Chapter 13, when Elena questions Lina about how Kostas looked, she lies that he didn't look "too bad." Lina neglects to tell her mother that her father had been beaten.