Gertrude is Kumalo's little sister—much littler, in fact, since she is twenty-five years and a whole generation younger than Kumalo. Before the beginning of the novel, Gertrude's husband left her with her small son to go work in the mines, and stopped writing letters home. Gertrude went to Johannesburg to look for him and disappeared in turn. When Gertrude stopped sending word back to Ndotsheni from the city, Absalom moved to Johannesburg to look for her—and we all know how that turned out (in disaster, that's how).
Not only is Gertrude the reason that Absalom goes to Johannesburg in the first place, but she is also the initial cause for Kumalo's trip to the city. Kumalo receives a letter from Msimangu asking him to come to the city and take care of his sick sister. It's only once Kumalo arrives that Msimangu breaks the news that Gertrude's "sickness" is her loose morality.
Gertrude is keeping company with a lot of guys—and there is a definite implication that she is a casual prostitute, since Msimangu tells Kumalo that, "These women [like Gertrude] sleep with any man for their price" (1.5.19).
But Msimangu's real objection is that she is "one of the queens, the liquor sellers" (1.6.9). In other words, Gertrude makes her money selling illegal, homemade alcohol—moonshine, bathtub gin, whatever you want to call it. And she's good at it, too, since she's one of the richest black women in Johannesburg. But her sales of illegal booze are corrupting both her and the people in her neighborhood: there's a lot of gambling that goes on in her house, and a man was killed there once.
Kumalo goes to Gertrude to confront her about her immorality, the shame she is bringing on her priest brother, and the danger her lifestyle is posing to her young son. Gertrude initially freaks out and promises to return to Ndotsheni with Kumalo. She sells all of her possessions and moves to Mrs. Lithebe's house with her brother to prepare for the move back to her hometown.
However, there are hints throughout Books 1 and 2 of the book that Gertrude is permanently ruined for village life and that she is too corrupted to go back to Ndotsheni (according to the moral logic of the book, at least). Mrs. Lithebe scolds her several times for her "careless" (2.27.10) talk and for her rowdy friends. And Mrs. Lithebe implies that Gertrude is a bad influence on Absalom's nameless (and personality-less) fiancée. Gertrude does struggle with her own sexual morality, even deciding briefly that she will become a nun. But when she finally runs away and abandons her young son with Kumalo, it seems like it was just a matter of time before she got sick of life with her much older, moralizing brother.
Gertrude's sexuality reveals something interesting (and kind of troubling) about Cry, the Beloved Country's view of women in general. For example, Gertrude and the rest of the family go to a church meeting to hear a black nun speaks about her experiences. This nun tells her church group that "God had taken from her that desire which is in the nature of women" (2.27.40). The idea that sexual desire is a sin specifically attached to women is a very old-fashioned religious belief, but it's one with which Gertrude, Msimangu, and maybe even the book as a whole seems to agree.
After hearing this nun's story, Gertrude confesses to Mrs. Lithebe: "I am a weak woman, you know it. I laugh and speak carelessly. Perhaps it would help me to become a nun" (2.27.49). Now, this whole bit about laughing and speaking carelessly doesn't seem so bad to us. But her laughter seems to be a euphemism, a symbol of her general sexual lack of restraint. And of course, Gertrude totally fails to become a nun, since she doesn't have the self-discipline for it.
Even her name indicates her excessive sexuality: probably the most famous Gertrude out there is Hamlet's mother. In the play Hamlet, Gertrude's super quick remarriage to her husband's potentially murderous brother after her husband's death is what makes Hamlet so deeply crazy. We know that Shakespeare is on Paton's mind, since Jarvis finds a book of Shakespeare's plays on his son's bookshelf (2.20.5). We don't think this overlap in naming between two women who have trouble controlling their desires can be totally accidental. So, even though Cry, the Beloved Country tells us that Gertrude's worst crime is her illegal booze dealing, it strongly implies that her real problem is that she likes sex too much.
We can contrast Gertrude's example with the other totally unredeemable member of the Kumalo family, John: we know that John lives with a woman who isn't his wife. But while Msimangu criticizes John for his lack of faithfulness, the main issue that the book keeps returning to in his case is the sin of pride and ambition. His sexuality doesn't receive the same criticism that Gertrude's "desire" (2.27.50) does. This difference implies a double standard for how men and women's sexual morality is treated in this book, with women taking most of the blame for lust.