You can introduce another useState() for writing into <input> element. And when the Submit button is pressed, you only push the someText to the group array.

const NewGroup = () => {
  const [group, setGroup] = useState([]);
  const [someText, setSomeText] = useState("");
  const addToGroup = (e) => {
    // Takes up all the present elements(of array group) and adds new element (search)
    let temp = [...group, someText];
    setGroup(temp);
  };
  return (
    <div>
      <input type="text" name="" id="" value={someText} onChange={(e) => { setSomeText(e.target.value); }} />
      <button onClick={addToGroup}>submit</button>
      <div>
        <select name="" id="">
          {group.map((category, index) => {
            return (
              <option key={index} value="">
                {category}
              </option>
            );
          })}
        </select>
      </div>
    </div>
  );
}

export default NewGroup;

Here's the link to the working app https://codesandbox.io/s/boring-ptolemy-0cwzmi?file=/src/App.js

First, you should not be getting the value of your input by using the DOM directly, instead you should be using a Ref.

Secondly, the issue is that you're mutating the group value directly instead of copying it. When you call setGroup, React doesn't realize there was any change (because the previous value equals the current value). Instead you want to copy the array and add your new element:

// Previously: const newGroup = group;
const newGroup = [...group];

That should be all you have to do to get your code to work. However, I simplified and cleaned it up a little more with inline comments below.

const NewGroup = () => {
    // Here's where our input will be stored so we can use it later (instead of querying the dom directly)
    const inputRef = React.useRef();
    const [group, setGroup] = React.useState([]);
    // Use React.useCallback so the function doesn't change on every render
    const addToGroup = React.useCallback((e) => {
        // We call setGroup with a function that returns the new group value. The first argument is the current group value. We return a new array by spreading the existing value and adding our new value
        setGroup((v) => [...v, inputRef.current.value]);
    }, []);
    return (
        <div>
            <input type="text" ref={inputRef} name="" id="" />
            <button onClick={addToGroup}>submit</button>
            <div>
                <select name="" id="">
                    {group.map((category) => {
                        return (
                            <option value="" key={category}>
                                {category}
                            </option>
                        );
                    })}
                </select>
            </div>
        </div>
    );
};