Both Buddhism and Jainism shared certain tenets and beliefs.

Option I proposes that both religions avoided extremities of penance and enjoyment. However, this is not entirely true. Jainism is well-known for its extreme penance and rigorous asceticism, while Buddhism advocates the `Middle Path` between self-indulgence and self-mortification.

Option II is correct in asserting that both Buddhism and Jainism showed indifference to the authority of the Vedas, setting them apart from the dominant Vedic Hinduism of their time.

Option III affirms that both religions denied the efficacy of rituals, promoting insight, morality, and meditation as paths to liberation rather than ritualistic observance.

Option IV is also correct, as both Buddhism and Jainism placed a profound emphasis on non-violence, which extended to all living beings, including animals.

Therefore, the correct answer is option 2 combining options II, III, and IV. It illustrates the shared beliefs of Buddhism and Jainism regarding the authority of Vedas, denial of rituals, and non-injury to animal life.