Step 1: Acne vulgaris is an androgen-driven disorder. Androgens, chiefly testosterone (and its potent skin metabolite dihydrotestosterone), stimulate the sebaceous glands.
Step 2: This androgenic stimulation increases sebum production and promotes hyperkeratinisation of the pilosebaceous duct. The trapped sebum and keratin form comedones, which then become colonised by Propionibacterium (Cutibacterium) acnes, leading to inflammation.
Step 3: This is why acne flares at puberty and in androgen-excess states such as PCOS, and why testosterone is the hormone responsible.
Step 4: Why not the others. Estrogen actually reduces sebum and improves acne. Thyroid hormone has no direct causal role. Gonadotropins act indirectly by stimulating gonadal steroid output, but the direct effector on the sebaceous gland is the androgen testosterone.