The correct answer is option 4: State shall own or control key industries and services. The Karachi resolution of the Indian National Congress in 1931 advocated for the state to have ownership or control over key industries and services. This resolution was an important step in the Congress` vision for economic development and self-reliance in India.

Option 1, which states that the state shall not own or control key industries and services, is not correct. This goes against the intent of the Karachi resolution which called for state control.

Option 2, which suggests that the state shall handover key industries and services to Indian business groups, is also not correct. The resolution did not mention handing over control to any specific group, but rather emphasized state ownership or control.

Option 3, which proposes allowing Indian business groups to invest fifty per cent of the capital, is not mentioned in the Karachi resolution. The resolution focused on state ownership or control, not on the percentage of investment by business groups.

Therefore, the correct answer is option 4, as it accurately reflects the key aspect advocated by the Karachi resolution of Congress in 1931.