Learn Finnish: Ryhmä, siirtykää eteenpäin.

It is the plural or formal imperative of siirtyä (to move or shift). To form it, take the infinitive stem (siirty-) and add the plural imperative suffix -kää. We use -kää instead of -kaa because of vowel harmony: the front vowels y and ä in the stem mean the ending must also use front vowels.

Why do we use siirtyä here for moving instead of another verb like liikkua?

Siirtyä specifically means moving or shifting from one place to another. Liikkua is more about general physical movement (like exercising or just not being still). Since the group is being told to change their location, siirtykää is the perfect choice.

What is the difference between eteen and eteenpäin?

Eteen means to the front (a specific destination), while eteenpäin means forward or onwards (a continuous direction). The suffix -päin adds that directional, ongoing feeling. So siirtykää eteenpäin means move onward or forward rather than just stepping to the front of something.

Is ryhmä in a special case because we are talking directly to the group?

No, Finnish just uses the basic dictionary form (nominative case) when addressing someone or something directly. There is no special calling case, so ryhmä stays exactly as it is.