Penelope's actions help develop the theme of loyalty in the Odyssey by demonstrating her steadfast commitment to her husband, Odysseus. By refusing to marry one of the suitors despite their pressure and taking control of her home, Penelope shows her loyalty to Odysseus and honors their marriage vows. Additionally, her loyalty is further exemplified by weaving a burial shroud for her father-in-law, Laertes, as she remains faithful to her duty to her husband's family. Finally, organizing an archery contest to test the suitors' abilities and potentially reveal Odysseus' identity shows Penelope's loyalty and devotion to him, as she believes in his return and wants to ensure that only he can fulfill the challenging task. Overall, Penelope's actions throughout the epic showcase her unwavering loyalty to Odysseus and contribute to the development of the theme of loyalty in the Odyssey.