Can a purchase order whose duration is 6 months be extended? There are no option clauses included .
The Government almost always includes FAR 52.217-8 Option to Extend Services in each service requirement for just this type of scenario. This isn't the standard option clause that is obvious because of option CLINS. But, you definitively stated that no option clauses were included.
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Without any of the option clauses, the only way to extend the contract is to make a scope determination and negotiate the cost of a bilateral modification for a four month increase. Here again, the Government's silence with regards to options in the contract will make it more challenging to develop a favorable scope determination.
I'm assuming, although your questions didn't state this, that this is likely a small dollar FAR 13 action. So you may just go directly to FAR 13.501, generate a sole source justification for a new purchase order to the incumbent, and chalk it up to a lesson learned.